{ "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", "
\n", " \"Cercha\n", "
\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", "
\n", " \"Cercha\n", "
\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", "
\n", " \"Cercha\n", "
\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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"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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"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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eK65Ty6ajIKpNFGTGBvNhzvuKjHrgawTftG0NOo/Feq5Utja5Q3l0eL9UZAQFU5/9usfUT78x6Uf+Zh2TWLKRKGP563zGcR9KBmNZHY0TujFTyPWU+lLijwHCbb8nA9Z7Hd+5y+r6TMT40dfrjj/646Yt5yhy/3x7Lnj/yuMlXXBNYaVjOA45fljt5uH0HG4Ko/i7cw8lg7GsdN6hODX9Pw5NF7dcT6foaiz8XXvUlOuz2Log8Hx94IvK/DxuDHoKMovOf2IGb7YpsKZEbUqzK8xU5k4vu6ninafzq+FNoQ2D+HPegWUwiNWBORXUa/RuVU+pKAn+0XRByBBcVcSDYe1aBY2uBX6aYKi9IdZuLZCfT3HWPwNev18Kk/HDhwtigDf2wTd8JH9OP5wMRrI6HCevW1O2VT2lrpT4R2OAi0Jy3sITpX1fTC/uOyP6RcL4pn36foMesKsLT7m4qPtrnZfNptM2N8zF+g+wf+kZVNvahz/nmAzyXwCmq1Ulmu+4UU+dDvK/vVv+sXVB8E9Nfy2fDn8lA9G2FCUL8vifz40GWPXg/RIuVfdbxUN4vGrnPUX6zmpB6TCmz7cWXF4f/px7aBkMYHVoTjUlKySI4Sg9pYpU+EfxEk7GtPQCTXwxEMGXcErn5/Njf472MSg6LNfBsQIjKz77PO3zxMxePPm01LZqHw8d2g2A0gtGjhWDXIscVKaRP+UUTAZO39pYGaf6/6TXM7GZrKdOFxt1NUb+uiccI9169n8UXnXMQ9pqqxukS4E7zA2tu2EMCC/ogI6RIdK9UDM2rnxW1rdD5Tg/G/Hh9vORF75MKlu1j7bT3ZI/3DhWfFBMC+2rTJA/ZRVMBmWmQVbG6aR3IR3zaeI0Wk+dPu6Kv9oblwF2ivpV29wwaN9DLRlIpWN4q16dL5ufXxBuzYgX8jBGpbp8XgpbtQ0PI38waR9OoaU8Mai9+MNF4bAyUNt7szoypyH/P+LUS0+d7u2ev9qbG+BzHdB/uukXJQT2m+6BCRJsfUAoHP/sE9TnQxqLthfL4d35CRjf+bJsVZ7uFUKtL1l5QMD4lurKSifyR21jFAlt50FDYIusS8Ex6OTPSUeXQV9WR+dUUrCOg756SjWJ8Od/Mvs/vNAOXhH/mMwS29IYMRvrSor7Rdu7itxXvryk0ulmwABjVPKRC0qnHB4Z4YWOEdI7bWkTRpaQpd/sZjPsOAfj3DZ0zRXf/QZBfxSPbMiZ2PiHVbVhrfwprDpMBjfUWlkZp6pq9Truq6dUtnf+2B7s05tk1gNGKhbCBGQQ6GND6Eeech2GY6nREDiKnqqdOIZ0Bd7CA7aQPoFnaiJTtZu83/QJWAv3QOBwejrZAMua35ZkGTc6JLDITt4vqzqyNztDKliiLE+kJertU+ccHLlOG0vl8bMn6rClDPqAaePb5/y5ysTOqaudbRyPpKeTDbAUgX7jSX2pe1emLmXrkz8HR65jLPvQHl/G+I5nVzzTON7QOC9CiXVfT8TiNOL8NpX+WPGTIouJ86Ipeg8vv/kId2AYui2lG+cQmECaMQxAmTkpJcaTPeCZ2daq459fiY9qGUrQU5QvKN/Rbr62Q6icpXUTMM7djLpKGMMuQtPzU2MctQcs2PQvP+wQG4vqNC012XGqZUPAOE/XA2M4nWFXDSkyjtoASyCsN/CqQzB4x8Gfzh3nWfaJgHE+sRi7ZwzHkut/XnKMozXAetoxbtWv2RsUkXsi4iXnIyqCBS2xkYBxbkTTO8MY9kY1umCqjKM1wJLUU/p4OySW9Q2rXNfC7fzM/qCIsbZQJmCcyzzGHBnDMdSGnZMk4yhfwslQ8lPDTy9uExP9w63er3tyPlW54Iu8tspTzzPO0yVsDKcz7KohZcZ4wH9T/NNtu1gsni/YeKksktNn1hZGtfYCTnU8UPTeMd98Y545bbTgCBjn6apgDKcz7KohUcZ/V7sze4QH/JciQ7nYxhAwnPcF3o/pzboNdIxHzEpo+UI8OqafOPSSji8QsGCHhWYCxrmZTd8cY9iX1PhyKTL+QziwuVnAiDEVeNPP0stgMq63Frkvd3+ldXuVFrxvpfN1jNCat3yKqFRH6HpHTRMb4xzQvyH6YAzr/79D+PUpmwJjtYG5DWru9dmFdqIN8nrpC/ae7E86fq+bZvIF3nF1qUmeKBhlDEn2wU1tLfQgYJx7QOooYgw7AM2QnSLjwyxHKeHhAefrC8+gD1ZFgIBxDkAZmGQMBwIbUXxLxrp2vhwlL+GSDmosL+TolsAzfu0an3Sbt2iccZ5O3RhOZ9hVQ2yMD+MBdwnG8o2AETACaxBwTmC2IHvUHrBu9LkiK521xjWg2TWMQBsB09U2OpbXRCDal3BSaIaf3NVLt80WSW+CZulGoEjAdLVIw/aHEIjWAKsRTJ7wIyCYSszIBz7YeV+REQ98VeObtq1B57FQz5XK1iZ2KI/O8JeKjJ5g2rOfrEH99BmTftjvqBk7Sb9fMF3tx2mRUnvW0ygNsDOMH72B1XFpEoY75iu/rcPNVI5hbHjSGNOaAVb9DGkjjzHQubHXcRaUh/H2RtmlHmNj7PrJWZx4iJuu9sM1e6m962mUBlhSwmOtGlc8Xx/4yi/GtTFIMMyg859Kwsg2hYfKqE1ndoWZxtzpZTdVvNd0YzdIcqarg3DNVzgFPT2fD8c8NbknGt5tHmQEUXJ+5vmAYe1aAY2uBc7B8/WGWLu1gIf8u0/V9Rm2lk1/Vhpjh48YjF0PqZuu9oC0bJHd62l0BljyupSxLa73UBKhM45PlFg0yNUy9N1+cd4rBjjoAbu6MLbFBd35vFHm9bbdR+mCCR2IibHrL0/T1f6sZi2Zip6uboAF7p6in0ZcEorSmTBBf28wuPx/K/OVjGPutQYKsxjPG5feaICVj/dLuFTdbxWP6vHeULj5a+wcDemD6WpRM+LaT0JPVzXAUmgM3pVibXF05eGJ/iDD2bgIO0ZVke4EVksreq1KuglKp2/4vT/WFqMa9ICVTv8vX92gi4P7ou5gvUpPPhi7k4jFwnT1hCOqvZT09GJNsjJy9Nv+6pTb99/4W6ge+/TQliFpn1XPY9WZG2wdY8QxpL9pH+NK4Cd19uFJlc26Fjh2gX+yal9yVp+r64nOaewO8ZWksDV2ZSmarpZ5xHKUop5ioFZdjlLXw/BxzWx5SG3xUGtLSOqfQMlnGNWvig84dmmUJ/NHn+bS6b4oLTmpY182P9+f4+rAiIeWwcTIl+oKlUslTW01dmE9MF0NcNlK71PQU7WBLlghvD47187qQRfG66QPFyNHYBt8qaayeK2MxS16r/xTcPyzYhb0ZCSNBduL5WilH//7nSvqy2eesQ5qfcmqC0AY31JdxfNT2jd2zdKUDpiuNuNZNSdFPb1YlWD5Yhjcj4KaDTkrGMpyqZsjJklcqSwz4e4q4tXmS0sqnX5bDDD/LHmXgdIph2dHeKFjDPI7bTH4/kVglp6VuJlhxzkY57aha674/jfGrpcMTVd7YVquUKp6uulqaILKhAqM3WGn+y6nslbznARMV+ekeey6pEs4f9lqaFt6wEjhmeKjDu+XchaMwNYETFe3lkCC1+9lgGWxb6vtjL8dElgsJ+9fVR3ZG7VQBcrzfcGh7CjS1JYoVmWbQxYAbZPHEsBj4denbW1sTFf7EJyvTJss5rvKqaa19bSXAdZN8TJqUp/o2g07IU1rbw5ZQMTk0awXxqaZzdo5qctik1EQQ4Wop2BwcoTSHyt+4ilJGcXoPemhbV+zPAxD11O6cQ6BCaQZwwCUmZNSYtzLA56Z36Dq+OfXCY9CJ+npyHKSd5TH+g3V1dNCp1haAwHj3ABmQLIxHABrZNHUGEftAQs2fc9+RluTyDC8TctJNp1j6QUCxrkAY+SuMRwJbsBpKTKO2gBLNqzr8KpDRnjHwZ/OHedZ9omAcT6xGLtnDMeS639ecoyjNcB62jE+mIVyGmejuSciXnI+2qK/LK0kBIzzdD0whtMZdtWQKuNoDbAE8pQ+3g7BZH3DKlddUKd2mgRYW4GtVuiYCcZ5utyN4XSGXTUkyTjKl3AylvzU8FOI2wRD/3Cr9+uenE9VLvgir63y1POM83QJG8PpDLtqSJlxdB6wYNOlwEI4fhGdNvlgVGsv4FQHnxXy3jHfdWMu/59tFR0tzzhPl7gxnM6wq4bUGcfoAWM4WXDdj+nFINNXiUfMamf5Yjs6pp849JKO1fJZwMdCMwHj3Mymb44x7EtqfLnkGWPEmCbMp9mZIRVV5L7c/ZXW5lVa8L6VzmeNamsLK40vY5TqiK2tW96P2BjnibpvDJe3HSkwVhuY26B/9+uzC+1EG+T10hfsPdmfdPxeN83ki9BykndUFqOMIbFJGYLQNxjnvqSayxnDZjZz5aTIeNPlKOcSTJ96JDw84HwN4T7nWJnhBIzzcGbVM4xhlcj8x1sy1rXz5SiLL+H4xhprKhRjtlj6/M1fr0a1hxdydEvgGb/WPo23MDMB4zwdqDGczrCrhrUZ63o1u6p7zO0qHjAvufipHwqMIOgcYxs60dKMgBEwAkcnIAOMbc0GElRZyLa++S+Oc/05/BPmHQAAAABJRU5ErkJggg==\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." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.1" }, "varInspector": { "cols": { "lenName": 16, "lenType": 16, "lenVar": 40 }, "kernels_config": { "python": { "delete_cmd_postfix": "", "delete_cmd_prefix": "del ", "library": "var_list.py", "varRefreshCmd": "print(var_dic_list())" }, "r": { "delete_cmd_postfix": ") ", "delete_cmd_prefix": "rm(", "library": "var_list.r", "varRefreshCmd": "cat(var_dic_list()) " } }, "types_to_exclude": [ "module", "function", "builtin_function_or_method", "instance", "_Feature" ], "window_display": false } }, "nbformat": 4, "nbformat_minor": 2 }