Mecs4.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
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     "slide_type": "slide"
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   },
   "source": [
    "# 4. Análise de Posição\n",
    "Professor Paulo Victor <br>\n",
    "CEFET/RJ Campus Angra dos Reis - Sala 6E\n",
    "<br>\n",
    "*Este material serve apenas como um guia para estudos e para as aulas, não substituindo a leitura dos livros indicados e é de uso exclusivo dos alunos do CEFET/RJ campus Angra dos Reis.* "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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    }
   },
   "source": [
    "Uma vez que um projeto de mecanismo tenha sido **sintetizado**, ele deve ser **analisado**. <br>\n",
    "O principal objetivo de uma análise cinemática é determinar as acelerações de todas as partes móveis do conjunto. **Forças Dinâmicas** são proporcionais à aceleração conforme a segunda lei de Newton. Precisamos conhecer as forças dinâmicas para calcularmos as **tensões** nos componentes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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    }
   },
   "source": [
    "![](figs4-0.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## 4.1 Sistemas de Coordenadas"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "* Sistema de coordenadas global ou absoluto\n",
    "* Sistema de coordenadas local"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](figs4-1.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## 4.2 Posição e Deslocamento"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Posição"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "A posição de um ponto no **plano** pode ser definida por meio de um **vetor posição**. <br>\n",
    "A escolha dos **eixos de referência** é arbritrária, selecionada para satisfazer o observador. <br>\n",
    "Um vetor de duas dimensões tem dois atributos, que podem ser expressos tento na forma *polar* quanto em *coordenadas cartesianas*. \n",
    "* forma polar\n",
    "    * fornece o módulo e o ângulo do vetor: $\\vec{p} = | R | @ \\angle \\theta$\n",
    "* forma cartesiana\n",
    "    * fornece os componentes X e Y do vetor: $\\vec{p} = R_x,R_y$\n",
    "Observa-se que:\n",
    "$$ R = \\sqrt{R_x^2 + R_y^2}$$\n",
    "$$ \\theta = arctan \\left( \\dfrac{R_y}{R_x} \\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Deslocamentos\n",
    "O vetor posição $\\vec{R}_{BA}$ define o deslocamento do ponto $B$ em relação ao ponto $A$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](figs4-3.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "fragment"
    }
   },
   "source": [
    "A posição de $B$ em consideração a $A$ é igual a posição absoluta de $B$ menos a posição absoluta de $A$\n",
    "$$\\vec{R}_{BA} = \\vec{R}_B - \\vec{R}_A$$\n",
    "Ou:\n",
    "$$\\vec{R}_{BA} = \\vec{R}_{BO} - \\vec{R}_{AO}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## 4.3 Translação, Rotação e Movimento Complexo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Translação\n",
    "Todos os pontos do corpo tem o mesmo deslocamento. A orientação angular é mantida.\n",
    "#### Rotação \n",
    "Diferentes pontos do corpo suportam difenrentes deslocamentos, portanto há uma diferença de deslocamento entre quaisquer dois pontos escolhidos\n",
    "#### Movimento complexo\n",
    "É a união dos dois movimentos anteriores\n",
    "\n",
    "#### Teoremas:\n",
    "**Teorema de Euler** <br>\n",
    "O deslocamento geral de um corpo rígido com um ponto fixo é igual a rotação relacionada a algum eixo.<br>\n",
    "**Teorema de Chasles** <br>\n",
    "Qualquer deslocamento de um corpo rígido é equivalente à soma da translação de qualquer ponto naquele corpo com rotação sobre um eixo por meio desse ponto."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## 4.4 Análise Gráfica da Posição de Mecanismos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Para qualquer mecanismo de um GDL, somente um parâmetro é necessário para definir a posição de todos os elos. O parâmetro usualmente escolhido é o ângulo do elo de entrada.<br>\n",
    "Os comprimentos dos elos $a$, $b$, $c$ e $d$ são conhecidos."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](figs4-4.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](figs4-5.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## 4.5 Análise Algébrica da Posição de Mecanismos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "A mesma ideia pode ser utilizada para se fazer a análise algébrica do mecanismo de 4 barras."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "theta2,a,b,c,d = symbols('theta2 a b c d')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\cos{\\left(\\theta_{2} \\right)}$"
      ],
      "text/plain": [
       "a*cos(theta2)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ax = a*cos(theta2)\n",
    "Ax"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\sin{\\left(\\theta_{2} \\right)}$"
      ],
      "text/plain": [
       "a*sin(theta2)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ay = a*sin(theta2)\n",
    "Ay"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Escrevendo as duas equações dos círculos com centros em $A$ e $O_4$ e com os respectivos raios $b$ e $c$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "Bx,By = symbols('B_x B_y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle b^{2} = \\left(B_{x} - a \\cos{\\left(\\theta_{2} \\right)}\\right)^{2} + \\left(B_{y} - a \\sin{\\left(\\theta_{2} \\right)}\\right)^{2}$"
      ],
      "text/plain": [
       "Eq(b**2, (B_x - a*cos(theta2))**2 + (B_y - a*sin(theta2))**2)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b2 = (Bx - Ax)**2 + (By - Ay)**2\n",
    "Eq1 = Eq(b**2,b2)\n",
    "Eq1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - b^{2} + \\left(B_{x} - a \\cos{\\left(\\theta_{2} \\right)}\\right)^{2} + \\left(B_{y} - a \\sin{\\left(\\theta_{2} \\right)}\\right)^{2}$"
      ],
      "text/plain": [
       "-b**2 + (B_x - a*cos(theta2))**2 + (B_y - a*sin(theta2))**2"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq1 = b2 - b**2\n",
    "Eq1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle c^{2} = B_{y}^{2} + \\left(B_{x} - d\\right)^{2}$"
      ],
      "text/plain": [
       "Eq(c**2, B_y**2 + (B_x - d)**2)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c2 = (Bx - d)**2 + By**2\n",
    "Eq2 = Eq(c**2,c2)\n",
    "Eq2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle B_{y}^{2} - c^{2} + \\left(B_{x} - d\\right)^{2}$"
      ],
      "text/plain": [
       "B_y**2 - c**2 + (B_x - d)**2"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq2 = c2 - c**2\n",
    "Eq2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - B_{y}^{2} - b^{2} + c^{2} - \\left(B_{x} - d\\right)^{2} + \\left(B_{x} - a \\cos{\\left(\\theta_{2} \\right)}\\right)^{2} + \\left(B_{y} - a \\sin{\\left(\\theta_{2} \\right)}\\right)^{2}$"
      ],
      "text/plain": [
       "-B_y**2 - b**2 + c**2 - (B_x - d)**2 + (B_x - a*cos(theta2))**2 + (B_y - a*sin(theta2))**2"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq3 = Eq1 - Eq2\n",
    "Eq3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - 2 B_{x} a \\cos{\\left(\\theta_{2} \\right)} + 2 B_{x} d - 2 B_{y} a \\sin{\\left(\\theta_{2} \\right)} + a^{2} \\sin^{2}{\\left(\\theta_{2} \\right)} + a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - b^{2} + c^{2} - d^{2}$"
      ],
      "text/plain": [
       "-2*B_x*a*cos(theta2) + 2*B_x*d - 2*B_y*a*sin(theta2) + a**2*sin(theta2)**2 + a**2*cos(theta2)**2 - b**2 + c**2 - d**2"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(Eq3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{- 2 B_{y} a \\sin{\\left(\\theta_{2} \\right)} + a^{2} - b^{2} + c^{2} - d^{2}}{2 \\left(a \\cos{\\left(\\theta_{2} \\right)} - d\\right)}$"
      ],
      "text/plain": [
       "(-2*B_y*a*sin(theta2) + a**2 - b**2 + c**2 - d**2)/(2*(a*cos(theta2) - d))"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solBx = solve(Eq3,Bx)[0]\n",
    "solBx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{4 B_{y}^{2} a^{2} \\sin^{2}{\\left(\\theta_{2} \\right)}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} + B_{y}^{2} - \\frac{4 B_{y} a^{3} \\sin{\\left(\\theta_{2} \\right)}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} + \\frac{4 B_{y} a b^{2} \\sin{\\left(\\theta_{2} \\right)}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} - \\frac{4 B_{y} a c^{2} \\sin{\\left(\\theta_{2} \\right)}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} + \\frac{4 B_{y} a d^{2} \\sin{\\left(\\theta_{2} \\right)}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} + \\frac{2 B_{y} a d \\sin{\\left(\\theta_{2} \\right)}}{a \\cos{\\left(\\theta_{2} \\right)} - d} + \\frac{a^{4}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} - \\frac{2 a^{2} b^{2}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} + \\frac{2 a^{2} c^{2}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} - \\frac{2 a^{2} d^{2}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} - \\frac{a^{2} d}{a \\cos{\\left(\\theta_{2} \\right)} - d} + \\frac{b^{4}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} - \\frac{2 b^{2} c^{2}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} + \\frac{2 b^{2} d^{2}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} + \\frac{b^{2} d}{a \\cos{\\left(\\theta_{2} \\right)} - d} + \\frac{c^{4}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} - \\frac{2 c^{2} d^{2}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} - \\frac{c^{2} d}{a \\cos{\\left(\\theta_{2} \\right)} - d} - c^{2} + \\frac{d^{4}}{4 a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 8 a d \\cos{\\left(\\theta_{2} \\right)} + 4 d^{2}} + \\frac{d^{3}}{a \\cos{\\left(\\theta_{2} \\right)} - d} + d^{2}$"
      ],
      "text/plain": [
       "4*B_y**2*a**2*sin(theta2)**2/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) + B_y**2 - 4*B_y*a**3*sin(theta2)/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) + 4*B_y*a*b**2*sin(theta2)/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) - 4*B_y*a*c**2*sin(theta2)/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) + 4*B_y*a*d**2*sin(theta2)/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) + 2*B_y*a*d*sin(theta2)/(a*cos(theta2) - d) + a**4/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) - 2*a**2*b**2/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) + 2*a**2*c**2/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) - 2*a**2*d**2/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) - a**2*d/(a*cos(theta2) - d) + b**4/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) - 2*b**2*c**2/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) + 2*b**2*d**2/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) + b**2*d/(a*cos(theta2) - d) + c**4/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) - 2*c**2*d**2/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) - c**2*d/(a*cos(theta2) - d) - c**2 + d**4/(4*a**2*cos(theta2)**2 - 8*a*d*cos(theta2) + 4*d**2) + d**3/(a*cos(theta2) - d) + d**2"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq4 = Eq2.subs(Bx,solBx).expand()\n",
    "Eq4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(a*(a**2 - 2*a*d*cos(theta2) - b**2 + c**2 + d**2)*sin(theta2) + (-a*cos(theta2) + d)*sqrt(-a**4 + 4*a**3*d*cos(theta2) + 2*a**2*b**2 + 2*a**2*c**2 + 4*a**2*d**2*sin(theta2)**2 - 6*a**2*d**2 - 4*a*b**2*d*cos(theta2) - 4*a*c**2*d*cos(theta2) + 4*a*d**3*cos(theta2) - b**4 + 2*b**2*c**2 + 2*b**2*d**2 - c**4 + 2*c**2*d**2 - d**4))/(2*(a**2 - 2*a*d*cos(theta2) + d**2)),\n",
       " (a*(a**2 - 2*a*d*cos(theta2) - b**2 + c**2 + d**2)*sin(theta2) + (a*cos(theta2) - d)*sqrt(-a**4 + 4*a**3*d*cos(theta2) + 2*a**2*b**2 + 2*a**2*c**2 + 4*a**2*d**2*sin(theta2)**2 - 6*a**2*d**2 - 4*a*b**2*d*cos(theta2) - 4*a*c**2*d*cos(theta2) + 4*a*d**3*cos(theta2) - b**4 + 2*b**2*c**2 + 2*b**2*d**2 - c**4 + 2*c**2*d**2 - d**4))/(2*(a**2 - 2*a*d*cos(theta2) + d**2))]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solBy = solve(Eq4,By)\n",
    "solBy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(solBy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{a \\left(a^{2} - 2 a d \\cos{\\left(\\theta_{2} \\right)} - b^{2} + c^{2} + d^{2}\\right) \\sin{\\left(\\theta_{2} \\right)} + \\left(- a \\cos{\\left(\\theta_{2} \\right)} + d\\right) \\sqrt{- a^{4} + 4 a^{3} d \\cos{\\left(\\theta_{2} \\right)} + 2 a^{2} b^{2} + 2 a^{2} c^{2} + 4 a^{2} d^{2} \\sin^{2}{\\left(\\theta_{2} \\right)} - 6 a^{2} d^{2} - 4 a b^{2} d \\cos{\\left(\\theta_{2} \\right)} - 4 a c^{2} d \\cos{\\left(\\theta_{2} \\right)} + 4 a d^{3} \\cos{\\left(\\theta_{2} \\right)} - b^{4} + 2 b^{2} c^{2} + 2 b^{2} d^{2} - c^{4} + 2 c^{2} d^{2} - d^{4}}}{2 \\left(a^{2} - 2 a d \\cos{\\left(\\theta_{2} \\right)} + d^{2}\\right)}$"
      ],
      "text/plain": [
       "(a*(a**2 - 2*a*d*cos(theta2) - b**2 + c**2 + d**2)*sin(theta2) + (-a*cos(theta2) + d)*sqrt(-a**4 + 4*a**3*d*cos(theta2) + 2*a**2*b**2 + 2*a**2*c**2 + 4*a**2*d**2*sin(theta2)**2 - 6*a**2*d**2 - 4*a*b**2*d*cos(theta2) - 4*a*c**2*d*cos(theta2) + 4*a*d**3*cos(theta2) - b**4 + 2*b**2*c**2 + 2*b**2*d**2 - c**4 + 2*c**2*d**2 - d**4))/(2*(a**2 - 2*a*d*cos(theta2) + d**2))"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "By1 = solBy[0]\n",
    "By1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{a \\left(a^{2} - 2 a d \\cos{\\left(\\theta_{2} \\right)} - b^{2} + c^{2} + d^{2}\\right) \\sin{\\left(\\theta_{2} \\right)} + \\left(a \\cos{\\left(\\theta_{2} \\right)} - d\\right) \\sqrt{- a^{4} + 4 a^{3} d \\cos{\\left(\\theta_{2} \\right)} + 2 a^{2} b^{2} + 2 a^{2} c^{2} + 4 a^{2} d^{2} \\sin^{2}{\\left(\\theta_{2} \\right)} - 6 a^{2} d^{2} - 4 a b^{2} d \\cos{\\left(\\theta_{2} \\right)} - 4 a c^{2} d \\cos{\\left(\\theta_{2} \\right)} + 4 a d^{3} \\cos{\\left(\\theta_{2} \\right)} - b^{4} + 2 b^{2} c^{2} + 2 b^{2} d^{2} - c^{4} + 2 c^{2} d^{2} - d^{4}}}{2 \\left(a^{2} - 2 a d \\cos{\\left(\\theta_{2} \\right)} + d^{2}\\right)}$"
      ],
      "text/plain": [
       "(a*(a**2 - 2*a*d*cos(theta2) - b**2 + c**2 + d**2)*sin(theta2) + (a*cos(theta2) - d)*sqrt(-a**4 + 4*a**3*d*cos(theta2) + 2*a**2*b**2 + 2*a**2*c**2 + 4*a**2*d**2*sin(theta2)**2 - 6*a**2*d**2 - 4*a*b**2*d*cos(theta2) - 4*a*c**2*d*cos(theta2) + 4*a*d**3*cos(theta2) - b**4 + 2*b**2*c**2 + 2*b**2*d**2 - c**4 + 2*c**2*d**2 - d**4))/(2*(a**2 - 2*a*d*cos(theta2) + d**2))"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "By2 = solBy[1]\n",
    "By2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{a^{2} - \\frac{a \\left(a \\left(a^{2} - 2 a d \\cos{\\left(\\theta_{2} \\right)} - b^{2} + c^{2} + d^{2}\\right) \\sin{\\left(\\theta_{2} \\right)} + \\left(- a \\cos{\\left(\\theta_{2} \\right)} + d\\right) \\sqrt{- a^{4} + 4 a^{3} d \\cos{\\left(\\theta_{2} \\right)} + 2 a^{2} b^{2} + 2 a^{2} c^{2} + 4 a^{2} d^{2} \\sin^{2}{\\left(\\theta_{2} \\right)} - 6 a^{2} d^{2} - 4 a b^{2} d \\cos{\\left(\\theta_{2} \\right)} - 4 a c^{2} d \\cos{\\left(\\theta_{2} \\right)} + 4 a d^{3} \\cos{\\left(\\theta_{2} \\right)} - b^{4} + 2 b^{2} c^{2} + 2 b^{2} d^{2} - c^{4} + 2 c^{2} d^{2} - d^{4}}\\right) \\sin{\\left(\\theta_{2} \\right)}}{a^{2} - 2 a d \\cos{\\left(\\theta_{2} \\right)} + d^{2}} - b^{2} + c^{2} - d^{2}}{2 \\left(a \\cos{\\left(\\theta_{2} \\right)} - d\\right)}$"
      ],
      "text/plain": [
       "(a**2 - a*(a*(a**2 - 2*a*d*cos(theta2) - b**2 + c**2 + d**2)*sin(theta2) + (-a*cos(theta2) + d)*sqrt(-a**4 + 4*a**3*d*cos(theta2) + 2*a**2*b**2 + 2*a**2*c**2 + 4*a**2*d**2*sin(theta2)**2 - 6*a**2*d**2 - 4*a*b**2*d*cos(theta2) - 4*a*c**2*d*cos(theta2) + 4*a*d**3*cos(theta2) - b**4 + 2*b**2*c**2 + 2*b**2*d**2 - c**4 + 2*c**2*d**2 - d**4))*sin(theta2)/(a**2 - 2*a*d*cos(theta2) + d**2) - b**2 + c**2 - d**2)/(2*(a*cos(theta2) - d))"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bx1 = solBx.subs(By,By1)\n",
    "Bx1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{a^{2} - \\frac{a \\left(a \\left(a^{2} - 2 a d \\cos{\\left(\\theta_{2} \\right)} - b^{2} + c^{2} + d^{2}\\right) \\sin{\\left(\\theta_{2} \\right)} + \\left(a \\cos{\\left(\\theta_{2} \\right)} - d\\right) \\sqrt{- a^{4} + 4 a^{3} d \\cos{\\left(\\theta_{2} \\right)} + 2 a^{2} b^{2} + 2 a^{2} c^{2} + 4 a^{2} d^{2} \\sin^{2}{\\left(\\theta_{2} \\right)} - 6 a^{2} d^{2} - 4 a b^{2} d \\cos{\\left(\\theta_{2} \\right)} - 4 a c^{2} d \\cos{\\left(\\theta_{2} \\right)} + 4 a d^{3} \\cos{\\left(\\theta_{2} \\right)} - b^{4} + 2 b^{2} c^{2} + 2 b^{2} d^{2} - c^{4} + 2 c^{2} d^{2} - d^{4}}\\right) \\sin{\\left(\\theta_{2} \\right)}}{a^{2} - 2 a d \\cos{\\left(\\theta_{2} \\right)} + d^{2}} - b^{2} + c^{2} - d^{2}}{2 \\left(a \\cos{\\left(\\theta_{2} \\right)} - d\\right)}$"
      ],
      "text/plain": [
       "(a**2 - a*(a*(a**2 - 2*a*d*cos(theta2) - b**2 + c**2 + d**2)*sin(theta2) + (a*cos(theta2) - d)*sqrt(-a**4 + 4*a**3*d*cos(theta2) + 2*a**2*b**2 + 2*a**2*c**2 + 4*a**2*d**2*sin(theta2)**2 - 6*a**2*d**2 - 4*a*b**2*d*cos(theta2) - 4*a*c**2*d*cos(theta2) + 4*a*d**3*cos(theta2) - b**4 + 2*b**2*c**2 + 2*b**2*d**2 - c**4 + 2*c**2*d**2 - d**4))*sin(theta2)/(a**2 - 2*a*d*cos(theta2) + d**2) - b**2 + c**2 - d**2)/(2*(a*cos(theta2) - d))"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bx2 = solBx.subs(By,By2)\n",
    "Bx2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "dic1 = {a:10,b:15,c:15,d:10,theta2:pi/6}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\pi}{6}$"
      ],
      "text/plain": [
       "pi/6"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dic1[theta2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 23.601702192173$"
      ],
      "text/plain": [
       "23.6017021921730"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bx1num = Bx1.subs(dic1).evalf()\n",
    "Bx1num"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 8.66025403784439, 23.6017021921730, 10.0000000000000]"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X1 = [0,Ax.subs(dic1).evalf(),Bx1.subs(dic1).evalf(),d.subs(dic1).evalf()]\n",
    "X1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 5.00000000000000, 6.32405704239262, 0]"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y1 = [0,Ay.subs(dic1).evalf(),By1.subs(dic1).evalf(),0]\n",
    "Y1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "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": [
    "plt.figure()\n",
    "plt.plot(X1,Y1,'-o')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "scrolled": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lista=[1,2,4,5]\n",
    "max(lista)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "min(lista)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(lista)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def plot4barras(an,bn,cn,dn,thetan):\n",
    "    dicn = {a:an,b:bn,c:cn,d:dn,theta2:thetan}\n",
    "    elos = [an,bn,cn,dn]\n",
    "    elo_maior = max(elos)\n",
    "    elo_menor = min(elos)\n",
    "    soma_elos = sum(elos)\n",
    "    Xn = [0,Ax.subs(dicn).evalf(),Bx1.subs(dicn).evalf(),d.subs(dicn).evalf()]\n",
    "    Yn = [0,Ay.subs(dicn).evalf(),By1.subs(dicn).evalf(),0]\n",
    "    plt.figure(dpi=150)\n",
    "    plt.plot(Xn,Yn,'-o')\n",
    "    plt.xlim([-20,30])\n",
    "    plt.ylim([-20,30])\n",
    "    plt.grid(True)\n",
    "    plt.gca().set_aspect('equal', adjustable='box')\n",
    "\n",
    "    if (elo_maior+elo_menor)<=(sum(elos)-(elo_maior+elo_menor)):\n",
    "        print('Grashof')\n",
    "    else:\n",
    "        print('Não-Grashof')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "scrolled": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Grashof\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot4barras(10,10,12,12,pi/4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "from ipywidgets import interact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Widget Javascript not detected.  It may not be installed or enabled properly. Reconnecting the current kernel may help.\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "d9883a8427b64dd9bf326c14ed594138"
      }
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.plot4barras(an, bn, cn, dn, thetan)>"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(plot4barras,an=(1,20),bn=(1,20),cn=(1,20),dn=(1,20),thetan=(0,4*pi.evalf()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "from matplotlib import animation\n",
    "from IPython.display import HTML"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "fig, ax = plt.subplots(figsize=(12,12))\n",
    "\n",
    "ax.set_xlim(( -20, 20))\n",
    "ax.set_ylim((-20, 20))\n",
    "ax.grid(True)\n",
    "plt.gca().set_aspect('equal', adjustable='box')\n",
    "line, = ax.plot([], [], 'o-')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "th = np.linspace(0,4*np.pi,720)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def animate(i):\n",
    "    dic2 = {a:9,b:12,c:15,d:13,theta2:th[i]}\n",
    "    X1 = [0,Ax.subs(dic2).evalf(),Bx1.subs(dic2).evalf(),d.subs(dic2).evalf()]\n",
    "    Y1 = [0,Ay.subs(dic2).evalf(),By1.subs(dic2).evalf(),0]\n",
    "    line.set_data(X1,Y1)\n",
    "    return (line,)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def init():\n",
    "    line.set_data([],[])\n",
    "    return (line,)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "anim = animation.FuncAnimation(fig, animate, init_func=init,frames=720, interval=20, blit=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
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       "  <source type=\"video/mp4\" src=\"data:video/mp4;base64,AAAAIGZ0eXBNNFYgAAACAE00ViBpc29taXNvMmF2YzEAAAAIZnJlZQAD+j9tZGF0AAACrgYF//+q\n",
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       "\">\n",
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       "</video>"
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     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
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    "HTML(anim.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "**Observação:**\n",
    "$$\\theta_3 = tan^{-1} \\left( \\dfrac{B_y - A_y}{B_x - A_x} \\right) $$\n",
    "$$\\theta_4 = tan^{-1} \\left( \\dfrac{B_y}{B_x - d} \\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Representação do laço de vetores nos mecanismos\n",
    "Essa aproximação tem algumas vantagens na síntese analítica de mecanismos. Também possui vantagens na análise de velocidades e acelerações dos mecanismos. <br>\n",
    "Esse laço termina em si mesmo fazendo com que a somatória dos vetores ao seu redor seja zero. Os comprimentos dos vetores são os comprimentos dos elos, todos conhecidos. A posição atual do mecanismo é definida pelo ângulo de entrada $\\theta_2$ por ser um mecanismo com 1 GDL. Queremos obter os ângulos desconehcidos $\\theta_3$ e $\\theta_4$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](figs4-6.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$ \\vec{R}_{AO_2} + \\vec{R}_{BA} - \\vec{R}_{BO_4} - \\vec{R}_{O_4O_2} = 0$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "#### Vetores como números complexos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](figs4-7.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Podemos escrever o vetor $\\vec{R}_A$ como:\n",
    "$$\\vec{R}_A = r cos \\theta + i r sen \\theta $$ \n",
    "* O número imaginário $i$ é usado como um operador, não como um valor\n",
    "* Cada multiplicação do vetor $\\vec{R}_A$ por $i$ resulta em uma rotação anti-horária de $90^{\\circ}$\n",
    "Uma vantagem de usar a notação dos números complexos para representar vetores planos é obter a **Identidade de Euler**: \n",
    "$$ e^{\\pm i \\theta} = cos \\theta \\pm i sen \\theta $$\n",
    "Dessa forma, podemos escrever $\\vec{R}_A$ como: \n",
    "$$\\vec{R}_A = r e^{i \\theta}$$\n",
    "Observe que:\n",
    "$$\\dfrac{d e^{i \\theta}}{d \\theta} = i e^{i \\theta} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "r,theta = symbols('r theta')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle r e^{i \\theta}$"
      ],
      "text/plain": [
       "r*exp(I*theta)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RA = r*E**(I*theta)\n",
    "RA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Para simplificar, poderíamos fazer a seguinte importação <br>\n",
    "```{ .py }\n",
    "from sympy import E,I\n",
    "```\n",
    "Como alternativa, podemos fazer da seguinte forma:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "scrolled": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log(E)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -1$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E**(I*pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -1$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle r e^{i \\theta}$"
      ],
      "text/plain": [
       "r*exp(I*theta)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RA = r*E**(I*theta)\n",
    "RA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle i r e^{i \\theta}$"
      ],
      "text/plain": [
       "I*r*exp(I*theta)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RA.diff(theta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "#### Equação vetorial em malha fechada nos mecanismos de quatro barras"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](figs4-6.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$ \\vec{R}_{AO_2} + \\vec{R}_{BA} - \\vec{R}_{BO_4} - \\vec{R}_{O_4O_2} = 0$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "a,b,c,d = symbols('a b c d')\n",
    "theta1,theta2,theta3,theta4 = symbols('theta1 theta2 theta3 theta4')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a e^{i \\theta_{2}} + b e^{i \\theta_{3}} - c e^{i \\theta_{4}} - d e^{i \\theta_{1}}$"
      ],
      "text/plain": [
       "a*exp(I*theta2) + b*exp(I*theta3) - c*exp(I*theta4) - d*exp(I*theta1)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RAO2 = a*E**(I*theta2)\n",
    "RBA = b*E**(I*theta3)\n",
    "RBO4 = c*E**(I*theta4)\n",
    "RO4O2 = d*E**(I*theta1)\n",
    "Eq1 = RAO2 + RBA - RBO4 - RO4O2\n",
    "Eq1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Observa-se que $\\theta_1=0$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a e^{i \\theta_{2}} + b e^{i \\theta_{3}} - c e^{i \\theta_{4}} - d$"
      ],
      "text/plain": [
       "a*exp(I*theta2) + b*exp(I*theta3) - c*exp(I*theta4) - d"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq1 = Eq1.subs(theta1,0)\n",
    "Eq1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Passaremos a equação da malha fechada para a forma cartesiana (trigonométrica), para isso reescreveremos a equação utilizando senos e cossenos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\left(i \\sin{\\left(\\theta_{2} \\right)} + \\cos{\\left(\\theta_{2} \\right)}\\right) + b \\left(i \\sin{\\left(\\theta_{3} \\right)} + \\cos{\\left(\\theta_{3} \\right)}\\right) - c \\left(i \\sin{\\left(\\theta_{4} \\right)} + \\cos{\\left(\\theta_{4} \\right)}\\right) - d$"
      ],
      "text/plain": [
       "a*(I*sin(theta2) + cos(theta2)) + b*(I*sin(theta3) + cos(theta3)) - c*(I*sin(theta4) + cos(theta4)) - d"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq1 = (Eq1).rewrite(sin)\n",
    "Eq1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Vamos expandir a Eq1:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle i a \\sin{\\left(\\theta_{2} \\right)} + a \\cos{\\left(\\theta_{2} \\right)} + i b \\sin{\\left(\\theta_{3} \\right)} + b \\cos{\\left(\\theta_{3} \\right)} - i c \\sin{\\left(\\theta_{4} \\right)} - c \\cos{\\left(\\theta_{4} \\right)} - d$"
      ],
      "text/plain": [
       "I*a*sin(theta2) + a*cos(theta2) + I*b*sin(theta3) + b*cos(theta3) - I*c*sin(theta4) - c*cos(theta4) - d"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq1 = Eq1.expand()\n",
    "Eq1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Vamos separar a parte real da imaginária dessa equação:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\cos{\\left(\\theta_{2} \\right)} + b \\cos{\\left(\\theta_{3} \\right)} - c \\cos{\\left(\\theta_{4} \\right)} - d$"
      ],
      "text/plain": [
       "a*cos(theta2) + b*cos(theta3) - c*cos(theta4) - d"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq_real = Eq1.subs(I,0)\n",
    "Eq_real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle i a \\sin{\\left(\\theta_{2} \\right)} + i b \\sin{\\left(\\theta_{3} \\right)} - i c \\sin{\\left(\\theta_{4} \\right)}$"
      ],
      "text/plain": [
       "I*a*sin(theta2) + I*b*sin(theta3) - I*c*sin(theta4)"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq_im = Eq1 - Eq_real\n",
    "Eq_im"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\sin{\\left(\\theta_{2} \\right)} + b \\sin{\\left(\\theta_{3} \\right)} - c \\sin{\\left(\\theta_{4} \\right)}$"
      ],
      "text/plain": [
       "a*sin(theta2) + b*sin(theta3) - c*sin(theta4)"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq_im = Eq_im.subs(I,1)\n",
    "Eq_im"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Agora, vamos passar os termos que contenham $b$ para o lado esquerdo da equação:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - a \\cos{\\left(\\theta_{2} \\right)} + c \\cos{\\left(\\theta_{4} \\right)} + d$"
      ],
      "text/plain": [
       "-a*cos(theta2) + c*cos(theta4) + d"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bcostheta3 = solve(Eq_real,b*cos(theta3))[0]\n",
    "bcostheta3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle b \\cos{\\left(\\theta_{3} \\right)} = - a \\cos{\\left(\\theta_{2} \\right)} + c \\cos{\\left(\\theta_{4} \\right)} + d$"
      ],
      "text/plain": [
       "Eq(b*cos(theta3), -a*cos(theta2) + c*cos(theta4) + d)"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(b*cos(theta3),bcostheta3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - a \\sin{\\left(\\theta_{2} \\right)} + c \\sin{\\left(\\theta_{4} \\right)}$"
      ],
      "text/plain": [
       "-a*sin(theta2) + c*sin(theta4)"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bsentheta3 = solve(Eq_im,b*sin(theta3))[0]\n",
    "bsentheta3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle b \\sin{\\left(\\theta_{3} \\right)} = - a \\sin{\\left(\\theta_{2} \\right)} + c \\sin{\\left(\\theta_{4} \\right)}$"
      ],
      "text/plain": [
       "Eq(b*sin(theta3), -a*sin(theta2) + c*sin(theta4))"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(b*sin(theta3),bsentheta3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Elevando essas duas equações ao quadrado e somando, eliminamos $\\theta_3$ das equações."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(- a \\sin{\\left(\\theta_{2} \\right)} + c \\sin{\\left(\\theta_{4} \\right)}\\right)^{2} + \\left(- a \\cos{\\left(\\theta_{2} \\right)} + c \\cos{\\left(\\theta_{4} \\right)} + d\\right)^{2}$"
      ],
      "text/plain": [
       "(-a*sin(theta2) + c*sin(theta4))**2 + (-a*cos(theta2) + c*cos(theta4) + d)**2"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b2 = bcostheta3**2 + bsentheta3**2\n",
    "b2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle b^{2} = \\left(- a \\sin{\\left(\\theta_{2} \\right)} + c \\sin{\\left(\\theta_{4} \\right)}\\right)^{2} + \\left(- a \\cos{\\left(\\theta_{2} \\right)} + c \\cos{\\left(\\theta_{4} \\right)} + d\\right)^{2}$"
      ],
      "text/plain": [
       "Eq(b**2, (-a*sin(theta2) + c*sin(theta4))**2 + (-a*cos(theta2) + c*cos(theta4) + d)**2)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(b**2,b2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Reorganizando: \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{2} \\sin^{2}{\\left(\\theta_{2} \\right)} + a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 2 a c \\sin{\\left(\\theta_{2} \\right)} \\sin{\\left(\\theta_{4} \\right)} - 2 a c \\cos{\\left(\\theta_{2} \\right)} \\cos{\\left(\\theta_{4} \\right)} - 2 a d \\cos{\\left(\\theta_{2} \\right)} - b^{2} + c^{2} \\sin^{2}{\\left(\\theta_{4} \\right)} + c^{2} \\cos^{2}{\\left(\\theta_{4} \\right)} + 2 c d \\cos{\\left(\\theta_{4} \\right)} + d^{2}$"
      ],
      "text/plain": [
       "a**2*sin(theta2)**2 + a**2*cos(theta2)**2 - 2*a*c*sin(theta2)*sin(theta4) - 2*a*c*cos(theta2)*cos(theta4) - 2*a*d*cos(theta2) - b**2 + c**2*sin(theta4)**2 + c**2*cos(theta4)**2 + 2*c*d*cos(theta4) + d**2"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq2 = b2.expand() - b**2\n",
    "Eq2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Precisaremos reescrever o $\\sin(\\theta_4)$ e o $\\cos(\\theta_4)$ como uma função da tangente de $\\theta_4$ para facilitar a resolução da equação."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 \\tan{\\left(\\frac{\\theta_{4}}{2} \\right)}}{\\tan^{2}{\\left(\\frac{\\theta_{4}}{2} \\right)} + 1}$"
      ],
      "text/plain": [
       "2*tan(theta4/2)/(tan(theta4/2)**2 + 1)"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sin(theta4).rewrite(tan)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1 - \\tan^{2}{\\left(\\frac{\\theta_{4}}{2} \\right)}}{\\tan^{2}{\\left(\\frac{\\theta_{4}}{2} \\right)} + 1}$"
      ],
      "text/plain": [
       "(1 - tan(theta4/2)**2)/(tan(theta4/2)**2 + 1)"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cos(theta4).rewrite(tan)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Essas identidades trigonométricas são conhecidas como *meia identidade dos ângulos*:\n",
    "$$ sen \\theta_4 = \\dfrac{2 tan \\left( \\dfrac{\\theta_4}{2} \\right)}{1 + tan^2 \\left( \\dfrac{\\theta_4}{2} \\right)} \\qquad cos \\theta_4 = \\dfrac{1 - tan^2 \\left( \\dfrac{\\theta_4}{2} \\right)}{1 + tan^2 \\left( \\dfrac{\\theta_4}{2} \\right)} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "mida = {sin(theta4):sin(theta4).rewrite(tan),\n",
    "        cos(theta4):cos(theta4).rewrite(tan)}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "scrolled": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{2} \\sin^{2}{\\left(\\theta_{2} \\right)} + a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - \\frac{2 a c \\left(1 - \\tan^{2}{\\left(\\frac{\\theta_{4}}{2} \\right)}\\right) \\cos{\\left(\\theta_{2} \\right)}}{\\tan^{2}{\\left(\\frac{\\theta_{4}}{2} \\right)} + 1} - \\frac{4 a c \\sin{\\left(\\theta_{2} \\right)} \\tan{\\left(\\frac{\\theta_{4}}{2} \\right)}}{\\tan^{2}{\\left(\\frac{\\theta_{4}}{2} \\right)} + 1} - 2 a d \\cos{\\left(\\theta_{2} \\right)} - b^{2} + \\frac{c^{2} \\left(1 - \\tan^{2}{\\left(\\frac{\\theta_{4}}{2} \\right)}\\right)^{2}}{\\left(\\tan^{2}{\\left(\\frac{\\theta_{4}}{2} \\right)} + 1\\right)^{2}} + \\frac{4 c^{2} \\tan^{2}{\\left(\\frac{\\theta_{4}}{2} \\right)}}{\\left(\\tan^{2}{\\left(\\frac{\\theta_{4}}{2} \\right)} + 1\\right)^{2}} + \\frac{2 c d \\left(1 - \\tan^{2}{\\left(\\frac{\\theta_{4}}{2} \\right)}\\right)}{\\tan^{2}{\\left(\\frac{\\theta_{4}}{2} \\right)} + 1} + d^{2}$"
      ],
      "text/plain": [
       "a**2*sin(theta2)**2 + a**2*cos(theta2)**2 - 2*a*c*(1 - tan(theta4/2)**2)*cos(theta2)/(tan(theta4/2)**2 + 1) - 4*a*c*sin(theta2)*tan(theta4/2)/(tan(theta4/2)**2 + 1) - 2*a*d*cos(theta2) - b**2 + c**2*(1 - tan(theta4/2)**2)**2/(tan(theta4/2)**2 + 1)**2 + 4*c**2*tan(theta4/2)**2/(tan(theta4/2)**2 + 1)**2 + 2*c*d*(1 - tan(theta4/2)**2)/(tan(theta4/2)**2 + 1) + d**2"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq3 = Eq2.subs(mida)\n",
    "Eq3 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Para facilitar a solução da equação, vamos substituir $\\tan \\left( \\dfrac{\\theta_4}{2} \\right)$ por $x$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{c^{2} \\cdot \\left(4 x^{2} + \\left(x^{2} - 1\\right)^{2}\\right) + 2 c \\left(x^{2} + 1\\right) \\left(- 2 a x \\sin{\\left(\\theta_{2} \\right)} + a \\left(x^{2} - 1\\right) \\cos{\\left(\\theta_{2} \\right)} - d \\left(x^{2} - 1\\right)\\right) + \\left(x^{2} + 1\\right)^{2} \\left(a^{2} - 2 a d \\cos{\\left(\\theta_{2} \\right)} - b^{2} + d^{2}\\right)}{\\left(x^{2} + 1\\right)^{2}}$"
      ],
      "text/plain": [
       "(c**2*(4*x**2 + (x**2 - 1)**2) + 2*c*(x**2 + 1)*(-2*a*x*sin(theta2) + a*(x**2 - 1)*cos(theta2) - d*(x**2 - 1)) + (x**2 + 1)**2*(a**2 - 2*a*d*cos(theta2) - b**2 + d**2))/(x**2 + 1)**2"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "Eq4 = Eq3.subs(tan(theta4/2),x)\n",
    "Eq4 = Eq4.simplify()\n",
    "Eq4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Agora, basta resolver para $x$ essa equação do segundo grau:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "sol = solve(Eq4,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 a c \\sin{\\left(\\theta_{2} \\right)} - \\sqrt{- a^{4} + 4 a^{3} d \\cos{\\left(\\theta_{2} \\right)} + 2 a^{2} b^{2} + 4 a^{2} c^{2} \\sin^{2}{\\left(\\theta_{2} \\right)} + 4 a^{2} c^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 2 a^{2} c^{2} - 4 a^{2} d^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 2 a^{2} d^{2} - 4 a b^{2} d \\cos{\\left(\\theta_{2} \\right)} - 4 a c^{2} d \\cos{\\left(\\theta_{2} \\right)} + 4 a d^{3} \\cos{\\left(\\theta_{2} \\right)} - b^{4} + 2 b^{2} c^{2} + 2 b^{2} d^{2} - c^{4} + 2 c^{2} d^{2} - d^{4}}}{a^{2} + 2 a c \\cos{\\left(\\theta_{2} \\right)} - 2 a d \\cos{\\left(\\theta_{2} \\right)} - b^{2} + c^{2} - 2 c d + d^{2}}$"
      ],
      "text/plain": [
       "(2*a*c*sin(theta2) - sqrt(-a**4 + 4*a**3*d*cos(theta2) + 2*a**2*b**2 + 4*a**2*c**2*sin(theta2)**2 + 4*a**2*c**2*cos(theta2)**2 - 2*a**2*c**2 - 4*a**2*d**2*cos(theta2)**2 - 2*a**2*d**2 - 4*a*b**2*d*cos(theta2) - 4*a*c**2*d*cos(theta2) + 4*a*d**3*cos(theta2) - b**4 + 2*b**2*c**2 + 2*b**2*d**2 - c**4 + 2*c**2*d**2 - d**4))/(a**2 + 2*a*c*cos(theta2) - 2*a*d*cos(theta2) - b**2 + c**2 - 2*c*d + d**2)"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 a c \\sin{\\left(\\theta_{2} \\right)} + \\sqrt{- a^{4} + 4 a^{3} d \\cos{\\left(\\theta_{2} \\right)} + 2 a^{2} b^{2} + 4 a^{2} c^{2} \\sin^{2}{\\left(\\theta_{2} \\right)} + 4 a^{2} c^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 2 a^{2} c^{2} - 4 a^{2} d^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 2 a^{2} d^{2} - 4 a b^{2} d \\cos{\\left(\\theta_{2} \\right)} - 4 a c^{2} d \\cos{\\left(\\theta_{2} \\right)} + 4 a d^{3} \\cos{\\left(\\theta_{2} \\right)} - b^{4} + 2 b^{2} c^{2} + 2 b^{2} d^{2} - c^{4} + 2 c^{2} d^{2} - d^{4}}}{a^{2} + 2 a c \\cos{\\left(\\theta_{2} \\right)} - 2 a d \\cos{\\left(\\theta_{2} \\right)} - b^{2} + c^{2} - 2 c d + d^{2}}$"
      ],
      "text/plain": [
       "(2*a*c*sin(theta2) + sqrt(-a**4 + 4*a**3*d*cos(theta2) + 2*a**2*b**2 + 4*a**2*c**2*sin(theta2)**2 + 4*a**2*c**2*cos(theta2)**2 - 2*a**2*c**2 - 4*a**2*d**2*cos(theta2)**2 - 2*a**2*d**2 - 4*a*b**2*d*cos(theta2) - 4*a*c**2*d*cos(theta2) + 4*a*d**3*cos(theta2) - b**4 + 2*b**2*c**2 + 2*b**2*d**2 - c**4 + 2*c**2*d**2 - d**4))/(a**2 + 2*a*c*cos(theta2) - 2*a*d*cos(theta2) - b**2 + c**2 - 2*c*d + d**2)"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "A solução de $\\theta_4$ para a malha aberta é: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "scrolled": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 \\operatorname{atan}{\\left(\\frac{2 a c \\sin{\\left(\\theta_{2} \\right)} - \\sqrt{- a^{4} + 4 a^{3} d \\cos{\\left(\\theta_{2} \\right)} + 2 a^{2} b^{2} + 4 a^{2} c^{2} \\sin^{2}{\\left(\\theta_{2} \\right)} + 4 a^{2} c^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 2 a^{2} c^{2} - 4 a^{2} d^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 2 a^{2} d^{2} - 4 a b^{2} d \\cos{\\left(\\theta_{2} \\right)} - 4 a c^{2} d \\cos{\\left(\\theta_{2} \\right)} + 4 a d^{3} \\cos{\\left(\\theta_{2} \\right)} - b^{4} + 2 b^{2} c^{2} + 2 b^{2} d^{2} - c^{4} + 2 c^{2} d^{2} - d^{4}}}{a^{2} + 2 a c \\cos{\\left(\\theta_{2} \\right)} - 2 a d \\cos{\\left(\\theta_{2} \\right)} - b^{2} + c^{2} - 2 c d + d^{2}} \\right)}$"
      ],
      "text/plain": [
       "2*atan((2*a*c*sin(theta2) - sqrt(-a**4 + 4*a**3*d*cos(theta2) + 2*a**2*b**2 + 4*a**2*c**2*sin(theta2)**2 + 4*a**2*c**2*cos(theta2)**2 - 2*a**2*c**2 - 4*a**2*d**2*cos(theta2)**2 - 2*a**2*d**2 - 4*a*b**2*d*cos(theta2) - 4*a*c**2*d*cos(theta2) + 4*a*d**3*cos(theta2) - b**4 + 2*b**2*c**2 + 2*b**2*d**2 - c**4 + 2*c**2*d**2 - d**4))/(a**2 + 2*a*c*cos(theta2) - 2*a*d*cos(theta2) - b**2 + c**2 - 2*c*d + d**2))"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta4_1 = 2*atan(sol[0])\n",
    "theta4_1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Para obter $\\theta_4$ para um dado $\\theta_2$, basta substituir os comprimentos dos elos e o $\\theta_2$ por valores numéricos. Faremos isso através de um dicionário."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{a: 10, b: 15, c: 15, d: 10, theta2: pi/6}"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dic1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0.435213273165934$"
      ],
      "text/plain": [
       "0.435213273165934"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta4_1.subs(dic1).evalf()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Para obter $\\theta_3$ devemos proceder de maneira similar..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\cos{\\left(\\theta_{2} \\right)} + b \\cos{\\left(\\theta_{3} \\right)} - d$"
      ],
      "text/plain": [
       "a*cos(theta2) + b*cos(theta3) - d"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ccostheta4 = solve(Eq_real,c*cos(theta4))[0]\n",
    "ccostheta4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\sin{\\left(\\theta_{2} \\right)} + b \\sin{\\left(\\theta_{3} \\right)}$"
      ],
      "text/plain": [
       "a*sin(theta2) + b*sin(theta3)"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "csintheta4 = solve(Eq_im,c*sin(theta4))[0]\n",
    "csintheta4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(a \\sin{\\left(\\theta_{2} \\right)} + b \\sin{\\left(\\theta_{3} \\right)}\\right)^{2} + \\left(a \\cos{\\left(\\theta_{2} \\right)} + b \\cos{\\left(\\theta_{3} \\right)} - d\\right)^{2}$"
      ],
      "text/plain": [
       "(a*sin(theta2) + b*sin(theta3))**2 + (a*cos(theta2) + b*cos(theta3) - d)**2"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c2 = ccostheta4**2 + csintheta4**2\n",
    "c2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{2} \\sin^{2}{\\left(\\theta_{2} \\right)} + a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} + 2 a b \\sin{\\left(\\theta_{2} \\right)} \\sin{\\left(\\theta_{3} \\right)} + 2 a b \\cos{\\left(\\theta_{2} \\right)} \\cos{\\left(\\theta_{3} \\right)} - 2 a d \\cos{\\left(\\theta_{2} \\right)} + b^{2} \\sin^{2}{\\left(\\theta_{3} \\right)} + b^{2} \\cos^{2}{\\left(\\theta_{3} \\right)} - 2 b d \\cos{\\left(\\theta_{3} \\right)} - c^{2} + d^{2}$"
      ],
      "text/plain": [
       "a**2*sin(theta2)**2 + a**2*cos(theta2)**2 + 2*a*b*sin(theta2)*sin(theta3) + 2*a*b*cos(theta2)*cos(theta3) - 2*a*d*cos(theta2) + b**2*sin(theta3)**2 + b**2*cos(theta3)**2 - 2*b*d*cos(theta3) - c**2 + d**2"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq5 = (c2 - c**2).expand()\n",
    "Eq5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [],
   "source": [
    "mida2 = {sin(theta3):sin(theta3).rewrite(tan),\n",
    "        cos(theta3):cos(theta3).rewrite(tan)}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{2} \\sin^{2}{\\left(\\theta_{2} \\right)} + a^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} + \\frac{2 a b \\left(1 - \\tan^{2}{\\left(\\frac{\\theta_{3}}{2} \\right)}\\right) \\cos{\\left(\\theta_{2} \\right)}}{\\tan^{2}{\\left(\\frac{\\theta_{3}}{2} \\right)} + 1} + \\frac{4 a b \\sin{\\left(\\theta_{2} \\right)} \\tan{\\left(\\frac{\\theta_{3}}{2} \\right)}}{\\tan^{2}{\\left(\\frac{\\theta_{3}}{2} \\right)} + 1} - 2 a d \\cos{\\left(\\theta_{2} \\right)} + \\frac{b^{2} \\left(1 - \\tan^{2}{\\left(\\frac{\\theta_{3}}{2} \\right)}\\right)^{2}}{\\left(\\tan^{2}{\\left(\\frac{\\theta_{3}}{2} \\right)} + 1\\right)^{2}} + \\frac{4 b^{2} \\tan^{2}{\\left(\\frac{\\theta_{3}}{2} \\right)}}{\\left(\\tan^{2}{\\left(\\frac{\\theta_{3}}{2} \\right)} + 1\\right)^{2}} - \\frac{2 b d \\left(1 - \\tan^{2}{\\left(\\frac{\\theta_{3}}{2} \\right)}\\right)}{\\tan^{2}{\\left(\\frac{\\theta_{3}}{2} \\right)} + 1} - c^{2} + d^{2}$"
      ],
      "text/plain": [
       "a**2*sin(theta2)**2 + a**2*cos(theta2)**2 + 2*a*b*(1 - tan(theta3/2)**2)*cos(theta2)/(tan(theta3/2)**2 + 1) + 4*a*b*sin(theta2)*tan(theta3/2)/(tan(theta3/2)**2 + 1) - 2*a*d*cos(theta2) + b**2*(1 - tan(theta3/2)**2)**2/(tan(theta3/2)**2 + 1)**2 + 4*b**2*tan(theta3/2)**2/(tan(theta3/2)**2 + 1)**2 - 2*b*d*(1 - tan(theta3/2)**2)/(tan(theta3/2)**2 + 1) - c**2 + d**2"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq6 = Eq5.subs(mida2)\n",
    "Eq6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{b^{2} \\cdot \\left(4 y^{2} + \\left(y^{2} - 1\\right)^{2}\\right) + 2 b \\left(y^{2} + 1\\right) \\left(2 a y \\sin{\\left(\\theta_{2} \\right)} - a \\left(y^{2} - 1\\right) \\cos{\\left(\\theta_{2} \\right)} + d \\left(y^{2} - 1\\right)\\right) + \\left(y^{2} + 1\\right)^{2} \\left(a^{2} - 2 a d \\cos{\\left(\\theta_{2} \\right)} - c^{2} + d^{2}\\right)}{\\left(y^{2} + 1\\right)^{2}}$"
      ],
      "text/plain": [
       "(b**2*(4*y**2 + (y**2 - 1)**2) + 2*b*(y**2 + 1)*(2*a*y*sin(theta2) - a*(y**2 - 1)*cos(theta2) + d*(y**2 - 1)) + (y**2 + 1)**2*(a**2 - 2*a*d*cos(theta2) - c**2 + d**2))/(y**2 + 1)**2"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = symbols('y')\n",
    "Eq7 = Eq6.subs(tan(theta3/2),y)\n",
    "Eq7 = Eq7.simplify()\n",
    "Eq7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-2*a*b*sin(theta2) - sqrt(-a**4 + 4*a**3*d*cos(theta2) + 4*a**2*b**2*sin(theta2)**2 + 4*a**2*b**2*cos(theta2)**2 - 2*a**2*b**2 + 2*a**2*c**2 - 4*a**2*d**2*cos(theta2)**2 - 2*a**2*d**2 - 4*a*b**2*d*cos(theta2) - 4*a*c**2*d*cos(theta2) + 4*a*d**3*cos(theta2) - b**4 + 2*b**2*c**2 + 2*b**2*d**2 - c**4 + 2*c**2*d**2 - d**4))/(a**2 - 2*a*b*cos(theta2) - 2*a*d*cos(theta2) + b**2 + 2*b*d - c**2 + d**2),\n",
       " (-2*a*b*sin(theta2) + sqrt(-a**4 + 4*a**3*d*cos(theta2) + 4*a**2*b**2*sin(theta2)**2 + 4*a**2*b**2*cos(theta2)**2 - 2*a**2*b**2 + 2*a**2*c**2 - 4*a**2*d**2*cos(theta2)**2 - 2*a**2*d**2 - 4*a*b**2*d*cos(theta2) - 4*a*c**2*d*cos(theta2) + 4*a*d**3*cos(theta2) - b**4 + 2*b**2*c**2 + 2*b**2*d**2 - c**4 + 2*c**2*d**2 - d**4))/(a**2 - 2*a*b*cos(theta2) - 2*a*d*cos(theta2) + b**2 + 2*b*d - c**2 + d**2)]"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol2 = solve(Eq7,y)\n",
    "sol2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 \\operatorname{atan}{\\left(\\frac{- 2 a b \\sin{\\left(\\theta_{2} \\right)} - \\sqrt{- a^{4} + 4 a^{3} d \\cos{\\left(\\theta_{2} \\right)} + 4 a^{2} b^{2} \\sin^{2}{\\left(\\theta_{2} \\right)} + 4 a^{2} b^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 2 a^{2} b^{2} + 2 a^{2} c^{2} - 4 a^{2} d^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 2 a^{2} d^{2} - 4 a b^{2} d \\cos{\\left(\\theta_{2} \\right)} - 4 a c^{2} d \\cos{\\left(\\theta_{2} \\right)} + 4 a d^{3} \\cos{\\left(\\theta_{2} \\right)} - b^{4} + 2 b^{2} c^{2} + 2 b^{2} d^{2} - c^{4} + 2 c^{2} d^{2} - d^{4}}}{a^{2} - 2 a b \\cos{\\left(\\theta_{2} \\right)} - 2 a d \\cos{\\left(\\theta_{2} \\right)} + b^{2} + 2 b d - c^{2} + d^{2}} \\right)}$"
      ],
      "text/plain": [
       "2*atan((-2*a*b*sin(theta2) - sqrt(-a**4 + 4*a**3*d*cos(theta2) + 4*a**2*b**2*sin(theta2)**2 + 4*a**2*b**2*cos(theta2)**2 - 2*a**2*b**2 + 2*a**2*c**2 - 4*a**2*d**2*cos(theta2)**2 - 2*a**2*d**2 - 4*a*b**2*d*cos(theta2) - 4*a*c**2*d*cos(theta2) + 4*a*d**3*cos(theta2) - b**4 + 2*b**2*c**2 + 2*b**2*d**2 - c**4 + 2*c**2*d**2 - d**4))/(a**2 - 2*a*b*cos(theta2) - 2*a*d*cos(theta2) + b**2 + 2*b*d - c**2 + d**2))"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta3_1 = 2*atan(sol2[0])\n",
    "theta3_1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -2.70637938042386$"
      ],
      "text/plain": [
       "-2.70637938042386"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta3_1.subs(dic1).evalf()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Caso queira copiar e colar a solução de $\\theta_3$ e $\\theta_4$ para usar em outro código, use o comando print():"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2*atan((-2*a*b*sin(theta2) - sqrt(-a**4 + 4*a**3*d*cos(theta2) + 4*a**2*b**2*sin(theta2)**2 + 4*a**2*b**2*cos(theta2)**2 - 2*a**2*b**2 + 2*a**2*c**2 - 4*a**2*d**2*cos(theta2)**2 - 2*a**2*d**2 - 4*a*b**2*d*cos(theta2) - 4*a*c**2*d*cos(theta2) + 4*a*d**3*cos(theta2) - b**4 + 2*b**2*c**2 + 2*b**2*d**2 - c**4 + 2*c**2*d**2 - d**4))/(a**2 - 2*a*b*cos(theta2) - 2*a*d*cos(theta2) + b**2 + 2*b*d - c**2 + d**2))\n"
     ]
    }
   ],
   "source": [
    "print(theta3_1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2*atan((2*a*c*sin(theta2) - sqrt(-a**4 + 4*a**3*d*cos(theta2) + 2*a**2*b**2 + 4*a**2*c**2*sin(theta2)**2 + 4*a**2*c**2*cos(theta2)**2 - 2*a**2*c**2 - 4*a**2*d**2*cos(theta2)**2 - 2*a**2*d**2 - 4*a*b**2*d*cos(theta2) - 4*a*c**2*d*cos(theta2) + 4*a*d**3*cos(theta2) - b**4 + 2*b**2*c**2 + 2*b**2*d**2 - c**4 + 2*c**2*d**2 - d**4))/(a**2 + 2*a*c*cos(theta2) - 2*a*d*cos(theta2) - b**2 + c**2 - 2*c*d + d**2))\n"
     ]
    }
   ],
   "source": [
    "print(theta4_1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2 \\operatorname{atan}{\\left(\\frac{- 2 a b \\sin{\\left(\\theta_{2} \\right)} - \\sqrt{- a^{4} + 4 a^{3} d \\cos{\\left(\\theta_{2} \\right)} + 4 a^{2} b^{2} \\sin^{2}{\\left(\\theta_{2} \\right)} + 4 a^{2} b^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 2 a^{2} b^{2} + 2 a^{2} c^{2} - 4 a^{2} d^{2} \\cos^{2}{\\left(\\theta_{2} \\right)} - 2 a^{2} d^{2} - 4 a b^{2} d \\cos{\\left(\\theta_{2} \\right)} - 4 a c^{2} d \\cos{\\left(\\theta_{2} \\right)} + 4 a d^{3} \\cos{\\left(\\theta_{2} \\right)} - b^{4} + 2 b^{2} c^{2} + 2 b^{2} d^{2} - c^{4} + 2 c^{2} d^{2} - d^{4}}}{a^{2} - 2 a b \\cos{\\left(\\theta_{2} \\right)} - 2 a d \\cos{\\left(\\theta_{2} \\right)} + b^{2} + 2 b d - c^{2} + d^{2}} \\right)}\n"
     ]
    }
   ],
   "source": [
    "print(latex(theta3_1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## 4.6 Solução para Análise de Posições no Mecanismo Biela-Manivela"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "A mesma análise feita para o vetor laço utilizada para a junta pinada de quatro barras pode ser aplicada ao mecanismo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](figs4-8.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\vec{R}_2 + \\vec{R}_3 - \\vec{R}_4 - \\vec{R}_1 = 0$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "R2 = a*E**(I*theta2)\n",
    "R3 = b*E**(I*theta3)\n",
    "R4 = c*E**(I*theta4)\n",
    "R1 = d*E**(I*theta1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a e^{i \\theta_{2}} - b e^{i \\theta_{3}} - c e^{i \\theta_{4}} - d e^{i \\theta_{1}}$"
      ],
      "text/plain": [
       "a*exp(I*theta2) - b*exp(I*theta3) - c*exp(I*theta4) - d*exp(I*theta1)"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = R2 - R3 - R4 - R1\n",
    "eq1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Observa-se que $\\theta_1 = 0$ e $\\theta_4 = 90^{\\circ}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a e^{i \\theta_{2}} - b e^{i \\theta_{3}} - i c - d$"
      ],
      "text/plain": [
       "a*exp(I*theta2) - b*exp(I*theta3) - I*c - d"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = eq1.subs(theta1,0)\n",
    "eq1 = eq1.subs(theta4,pi/2)\n",
    "eq1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle i a \\sin{\\left(\\theta_{2} \\right)} + a \\cos{\\left(\\theta_{2} \\right)} - i b \\sin{\\left(\\theta_{3} \\right)} - b \\cos{\\left(\\theta_{3} \\right)} - i c - d$"
      ],
      "text/plain": [
       "I*a*sin(theta2) + a*cos(theta2) - I*b*sin(theta3) - b*cos(theta3) - I*c - d"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq2 = (eq1).rewrite(sin).expand()\n",
    "eq2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\cos{\\left(\\theta_{2} \\right)} - b \\cos{\\left(\\theta_{3} \\right)} - d$"
      ],
      "text/plain": [
       "a*cos(theta2) - b*cos(theta3) - d"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq2_real = eq2.subs(I,0)\n",
    "eq2_real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\sin{\\left(\\theta_{2} \\right)} - b \\sin{\\left(\\theta_{3} \\right)} - c$"
      ],
      "text/plain": [
       "a*sin(theta2) - b*sin(theta3) - c"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq2_im = (eq2 - eq2_real).subs(I,1)\n",
    "eq2_im"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[pi - asin((a*sin(theta2) - c)/b), asin((a*sin(theta2) - c)/b)]"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "th3 = solve(eq2_im,theta3)\n",
    "th3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Note que existem novamente duas soluções correspondentes aos dois circuitos do mecanismo. A função *arco seno* possui duas soluções."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\operatorname{asin}{\\left(\\frac{a \\sin{\\left(\\theta_{2} \\right)} - c}{b} \\right)}$"
      ],
      "text/plain": [
       "asin((a*sin(theta2) - c)/b)"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "th3 = th3[1]\n",
    "th3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[a*cos(theta2) - b*cos(theta3)]"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsol = solve(eq2_real,d)\n",
    "dsol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\cos{\\left(\\theta_{2} \\right)} - b \\cos{\\left(\\theta_{3} \\right)}$"
      ],
      "text/plain": [
       "a*cos(theta2) - b*cos(theta3)"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsol = dsol[0]\n",
    "dsol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\cos{\\left(\\theta_{2} \\right)} - b \\sqrt{1 - \\frac{\\left(a \\sin{\\left(\\theta_{2} \\right)} - c\\right)^{2}}{b^{2}}}$"
      ],
      "text/plain": [
       "a*cos(theta2) - b*sqrt(1 - (a*sin(theta2) - c)**2/b**2)"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsol = dsol.subs(theta3,th3)\n",
    "dsol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## 4.7 Ângulos de Transmissão"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "O ângulo de transmissão $\\mu$ é definido como o *ângulo entre o elo de saída e o acoplador*. <br>\n",
    "Iremos expandir essa definição para representar o ângulo entre quaisquer elos em um mecanismo.  -> O ângulo entre qualquer elo de saída e o acoplador que o movimenta é um ângulo de transmissão. \n",
    "<br>\n",
    "Agora que desenvolvemos as expressões analíticas para os ângulos de todos os elos do mecanismo, fica fácil determinar algebricamente o ângulo de transmissão. Ele é apenas a diferença entre os ângulos dos dois elos.<br>\n",
    "Por convenção, tomamos o valor absoluto da diferença e o forçamos a ser um ângulo agudo. \n",
    "$$\\theta_{trans} = | \\theta_3 - \\theta_4 |$$\n",
    "se $\\theta_{trans} > \\pi/2$ então $\\mu = \\pi - \\theta_{trans}$ caso contrário $\\mu = \\theta_{trans}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def ang_trans(th3,th4):\n",
    "    th_trans = abs(th3 - th4)\n",
    "    pi = 3.14159265358979\n",
    "    if th_trans > (pi)/2:\n",
    "        return pi - th_trans\n",
    "    else:\n",
    "        return th_trans"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Esse cálculo pode ser feito para qualquer junta de um mecanismo, utilizando os ângulos dos elos apropriados."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Valores extremos do ângulo de transmissão\n",
    "Para um mecanismo manivela de quatro barras de Grashof, o valor mínimo do ângulo de transmissão ocorrerá quando a manivela for colinear ao elo terra. Os valores do ângulo de transmissão nessas posições são facilmente calculados pela lei dos cossenos. <br>\n",
    "Dependendo da geometria do mecanismo, o valor mínimo de transmissão $\\mu_{min}$ poderá ocorrer quando os elos 1 e 2 forem colineares e sobrepostos, ou quando os elos 1 e 2 forem colineares e não sobrepostos."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](figs4-9.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Para o caso sobreposto: \n",
    "$$\\mu_1 = \\gamma_1 = arccos \\left(\\dfrac{b^2 + c^2 - (d+a)^2}{2bc} \\right) $$\n",
    "Para o caso estendido: \n",
    "$$ \\mu_2 = \\gamma_2 = arccos \\left( \\dfrac{b^2 + c^2 - (d+a)^2}{2bc} \\right) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "O ângulo mínimo de transmissão $\\mu_{min}$ em um mecanimos manivela seguidor de Grashof é o menor dos valores de $\\mu_1$ e $\\mu_2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Para um mecanismo **duplo seguidor de Grashof**, o ângulo de transmissão pode variar de $0^\\circ$ a $90^\\circ$ porque o acoplador pode fazer uma revolução completa em relação aos outros elos. <br>\n",
    "Para um mecanismo **triplo seguidor não Grashof**, o ângulo de transmissão será $0^\\circ$ nos pontos de singularidade que ocorrem quando o seguidor de saída $c$ e o acoplador $b$ forem colinerares. Nos outros pontos de singularidade, quando o seguidor de entrada $a$ e o acoplador $b$ forem colineares, o ângulo de transmissão pode ser calculado pela lei dos cossenos: <br>\n",
    "quando: $\\qquad \\nu=0,$\n",
    "$$ \\mu = arccos \\left( \\dfrac{(a+b)^{2} + c^2 - d^2}{2c(a+b)} \\right) $$\n",
    "Esse não é o menor valor que o ângulo de transmissão $\\mu$ pode obter em um triplo seguidor, já que ele esse valor é zero."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](figs4-10.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " [Voltar para o capítulo anterior](Mecs3.ipynb)     $\\qquad$   [Avançar para o próximo capítulo](Mecs5.ipynb)"
   ]
  }
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\n",
          "text/plain": "<Figure size 750x750 with 1 Axes>"
         },
         "metadata": {
          "needs_background": "light"
         },
         "output_type": "display_data"
        }
       ]
      }
     },
     "1910a953ed3b4d6e8bce920ee7b0fb98": {
      "model_module": "@jupyter-widgets/base",
      "model_module_version": "1.2.0",
      "model_name": "LayoutModel",
      "state": {}
     },
     "1a9d1b441469480ca81c917f85276ec9": {
      "model_module": "@jupyter-widgets/base",
      "model_module_version": "1.2.0",
      "model_name": "LayoutModel",
      "state": {}
     },
     "1e7ba5f98b3f4c03ae5e181715520e99": {
      "model_module": "@jupyter-widgets/base",
      "model_module_version": "1.2.0",
      "model_name": "LayoutModel",
      "state": {}
     },
     "2138628862484b16979c45940d5ad8fc": {
      "model_module": "@jupyter-widgets/base",
      "model_module_version": "1.2.0",
      "model_name": "LayoutModel",
      "state": {}
     },
     "21d9c60f5f1b482ca0e0c4f196874cae": {
      "model_module": "@jupyter-widgets/controls",
      "model_module_version": "1.5.0",
      "model_name": "IntSliderModel",
      "state": {
       "description": "an",
       "layout": "IPY_MODEL_c6342a220cd14dc99369ed3afb27c884",
       "max": 20,
       "min": 1,
       "style": "IPY_MODEL_f12793bc35c9412ba6966ea883e46a4c",
       "value": 9
      }
     },
     "2a77e0af97d543cd91fb5c7253394cd9": {
      "model_module": "@jupyter-widgets/output",
      "model_module_version": "1.0.0",
      "model_name": "OutputModel",
      "state": {
       "layout": "IPY_MODEL_d78bede0c0e143669f38ff60beb58789",
       "outputs": [
        {
         "name": "stdout",
         "output_type": "stream",
         "text": "Grashof\n"
        },
        {
         "data": {
          "image/png": 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       "outputs": [
        {
         "name": "stdout",
         "output_type": "stream",
         "text": "Grashof\n"
        },
        {
         "data": {
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        "IPY_MODEL_eb8ed039d0e5485b97a7f61d1b9d00fe"
       ],
       "layout": "IPY_MODEL_ef3c5d4186034665b8916b322a4e4a8c"
      }
     },
     "919420f4784a41a3be95572623b50bd6": {
      "model_module": "@jupyter-widgets/output",
      "model_module_version": "1.0.0",
      "model_name": "OutputModel",
      "state": {
       "layout": "IPY_MODEL_65d638082d0a4da2b5b0291a9be34dd4",
       "outputs": [
        {
         "name": "stdout",
         "output_type": "stream",
         "text": "Não-Grashof\n"
        },
        {
         "data": {
          "image/png": 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