{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "###### Content under Creative Commons Attribution license CC-BY 4.0, code under MIT license (c)2014 L.A. Barba, C. Cooper, G.F. Forsyth, A. Krishnan." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Phugoid Motion" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Welcome to [**\"Practical Numerical Methods with Python!\"**](http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about) This course is a collaborative, online, open education project, where we aim to give a foundation in scientific computing. The focus is on numerical solution of problems modeled by ordinary and partial differential equations.\n", "\n", "This Jupyter Notebook introduces the problem we'll be studying in the **first module** of the course: the _phugoid model of glider flight_. We'll start with some background, explaining the physics, and working out the mathematical model. \n", "\n", "First, we'll look at an idealized motion where there is no drag, resulting in a simple harmonic motion. We can plot some interesting trajectories that will pique your imagination. In the next notebook, you'll learn to numerically integrate the differential equation using Euler's method. But hang on ... first things first. \n", "\n", "The term \"phugoid\" is used in aeronautics to refer to a motion pattern where an aircraft oscillates up and down —nose-up and climb, then nose-down and descend— around an equilibrium trajectory. The aircraft oscillates in altitude, speed and pitch, with only small (neglected) variations in the angle of attack, as it repeatedly exchanges kinetic and potential energy.\n", "\n", "A low-amplitude phugoid motion can be just a nuisance, as the aircraft does not exceed the stall angle of attack and nothing bad happens. But the mode can also be unstable leading to a stall or even a loop!\n", "\n", "Look at this video showing a Cessna single-engine airplane in phugoid motion:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/jpeg": 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\n", "text/html": [ "\n", " \n", " " ], "text/plain": [ "" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.display import YouTubeVideo\n", "YouTubeVideo('ysdU4mnRYdM')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "That doesn't look too good! What's happening? \n", "\n", "It can get a lot worse when an aircraft enters one of these modes that is unstable. For example, one of [NASA's Helios Solar Powered Aircraft](http://www.nasa.gov/centers/dryden/history/pastprojects/Helios/) prototype broke up in mid air due to extreme phugoid oscillations!\n", "\n", "Helios was a proof-of-concept solar electric-powered flying wing that broke the world altitude record for a non-rocket-powered aircraft in August 2001. But in June 26, 2003, it broke something else. The aircraft entered phugoid motion after encountering turbulence near the Hawaiian Island of Kauai. The high speed in the oscillatory movement exceeded the design limits, and it ended up wrecked in the Pacific Ocean. Luckily, the Helios was remotely operated, and nobody got hurt." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The physics of phugoids" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The phugoid oscillation has the aircraft pitching up and down, as it decelerates and accelerates. The trajectory might look like a sinusoid, as in the figure below. The assumption is that the forward velocity of the aircraft, $v$, varies in such a way that the angle of attack remains (nearly) constant, which means that we can assume a constant lift coefficient." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "![Image](./figures/oscillatory_trajectory.png)\n", "#### Figure 1. Trajectory of an aircraft in phugoid motion." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the descending portion of the trajectory, the aircraft's velocity increases as it proceeds from a peak to the minimum height—gaining kinetic energy at the expense of potential energy. The contrary happens in the upward segment, as its velocity decreases there.\n", "\n", "We measure the pitch angle (between the aircraft's longitudinal axis and the horizontal) as positive when the aircraft's nose is pointing up. In the portion of the trajectory below the center-line, where it curves upwards, the pitch angle $\\theta$ is increasing: $\\dot{\\theta}>0$. And where the trajectory curves down, the pitch angle is decreasing: $\\dot{\\theta}<0$, as shown in the figure.\n", "\n", "Let's remind ourselves of the forces affecting an aircraft in a downward glide. Look at the figure below: we show the flight path, the forces on the glider (no thrust), and the _glide angle_ or flight path angle, $\\gamma$, between the flight path and the horizontal." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "![Image](./figures/glider_forces.png)\n", "#### Figure 2. Forces on a glider." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The force of lift, $L$ —created by the airflow around the wings— is perpendicular to the trajectory, and the force of drag, $D$, is parallel to the trajectory. Both forces are expressed in terms of coefficients of lift and drag, $C_L$ and $C_D$, respectively, that depend on the wing design and _angle of attack_—the angle between the wing chord and the flight path.\n", "\n", "If you are not familiar with airplane aerodynamics, you might be getting confused with some terms here ... and all those angles! But be patient and look things up, if you need to. We're giving you a quick summary here.\n", "\n", "Lift and drag are proportional to a surface area, $S$, and the dynamic pressure: $1/2 \\rho v^2$, where $\\rho$ is the density of air, and $v$ the forward velocity of the aircraft. The equations for lift and drag are:\n", "\n", "$$\n", "\\begin{eqnarray}\n", "L &=& C_L S \\times \\frac{1}{2} \\rho v^2 \\\\\n", "D &=& C_D S \\times \\frac{1}{2} \\rho v^2\n", "\\end{eqnarray}\n", "$$\n", "\n", "If the glider were in equilibrium, the forces would balance each other. We can equate the forces in the directions perpendicular and parallel to the trajectory, as follows:\n", "\n", "$$\n", "\\begin{equation}\n", "L = W \\cos \\gamma \\quad \\text{and} \\quad D = W \\sin \\gamma\n", "\\end{equation}\n", "$$\n", "\n", "where $W$ represents the weight of the glider.\n", "\n", "In the figure, we've drawn the angle $\\gamma$ as the _glide angle_, formed between the direction of motion and the horizontal. We are not bothered with the _sign_ of the angle, because we draw a free-body diagram and take the direction of the forces into account in writing our balance equations. But later on, we will need to be careful with the sign of the angles. It can cause you a real headache to keep this straight, so be patient!\n", "\n", "It looks like we've set this up to do a little bit of mathematics. Are you ready?\n", "\n", "But before, a short glimpse of the history." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Lanchester's Aerodonetics" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\"Phugoid theory\" was first described by the British engineer Frederick W. Lanchester in _\"Aerodonetics\"_ (1909). This book is so old that it is now in the public domain, so you can actually download [from Google Books](http://books.google.com/books?id=6hxDAAAAIAAJ&dq=%22phugoid%20theory%20deals%20with%20the%20longitudinal%20stability%22&pg=PA37#v=onepage&q=%22phugoid%20theory%20deals%20with%20the%20longitudinal%20stability%22&f=false) a PDF file of a scan, or read it online. \n", "\n", "Lanchester defines phugoid theory as the study of longitudinal stability of a flying machine (aerodone). He first considered the simplification where drag and moment of inertia are neglected. Then he included these effects, obtaining an equation of stability. In addition to describing many experiments by himself and others, Lanchester also reports on _\"numerical work ... done by the aid of an ordinary 25-cm slide rule.\"_ Go figure!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Ideal case of zero drag" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this section, we follow the derivation given by Milne-Thompson (1966), which we find a little bit easier than that of the original in \"Aerodonetics\"!\n", "\n", "An aircraft flying in a steady, straight horizontal flight has a lift equal to its weight. The velocity in this condition is sometimes called _trim velocity_ (\"trim\" is what pilots do to set the controls to just stay in a steady flight). Let's use $v_t$ for the trim velocity, and from $L=W$ deduce that:\n", "\n", "$$\n", "\\begin{equation}\n", "W = C_L S \\times\\frac{1}{2} \\rho v_t^2\n", "\\end{equation}\n", "$$\n", "\n", "The weight $W$ is constant for the aircraft, but the lift at any other flight condition depends on the flight speed, $v$. We can use the expression for the weight in terms of $v_t$ to obtain the ratio $L/W$ at any other flight velocity, as follows:\n", "\n", "$$\n", "\\begin{equation}\n", "\\frac{L}{W}= \\frac{v^2}{v_t^2}\n", "\\end{equation}\n", "$$\n", "\n", "Imagine that the aircraft experienced a little upset, a wind gust, and it finds itself off the \"trim\" level, in a curved path with an instantaneous angle $\\theta$. In the sketch below, we exaggerate the curved trajectory of flight to help you visualize what we'll do next. The angle $\\theta$ (using the same name as Milne-Thompson) is between the _trajectory_ and the horizontal, positive up." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "#### Figure 3. Curved trajectory of the aircraft going up." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can form a free body diagram to determine the balance of forces. \n", "\n", "\n", "\n", "#### Figure 4. Free body diagram of the aircraft trajectory" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From the free body diagram, we can see that\n", "\n", "$$\n", "\\begin{equation}\n", "\\vec{L} + \\vec{W} = m\\vec{a} = \\frac{mv^2}{R}\\hat{n} + m \\frac{dv}{dt}\\hat{t}\n", "\\end{equation}\n", "$$\n", "\n", "where $\\frac{v^2}{R}$ is the centripetal acceleration and $R$ is the radius of curvature of the trajectory.\n", "If we decompose the lift and weight into their normal and tangential components we get\n", "\n", "$$\n", "\\begin{equation}\n", "L\\hat{n} + W_n\\hat{n} + W_t\\hat{t} = \\frac{mv^2}{R}\\hat{n} + m \\frac{dv}{dt}\\hat{t}\n", "\\end{equation}\n", "$$\n", "\n", "The component of the weight in the normal direction ($W_n$) is\n", "\n", "$$\n", "\\begin{equation}\n", "W_n = -W \\cos \\theta\n", "\\end{equation}\n", "$$\n", "\n", "If we then consider that all of the components in $\\hat{n}$ must balance out, we arrive at\n", "\n", "$$\n", "\\begin{equation}\n", "L - W \\cos \\theta = \\frac{mv^2}{R}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can rewrite this as\n", "\n", "$$\n", "\\begin{equation}\n", "L- W \\cos \\theta = \\frac{W}{g} \\frac{v^2}{R}\n", "\\end{equation}\n", "$$\n", "\n", "where $g$ is the acceleration due to gravity. Rearrange this by dividing the equation by the weight, and use the expression we found for $L/W$, above. The following equation results:\n", "\n", "$$\n", "\\begin{equation}\n", "\\frac{v^2}{v_t^2}-\\cos \\theta = \\frac{v^2}{g R}\n", "\\end{equation}\n", "$$\n", "\n", "Recall that we simplified the problem assuming that there is no friction, which means that the total energy is constant (the lift does no work). If $z$ represents the depth below a reference horizontal line, the energy per unit mass is (kinetic plus potential energy):\n", "\n", "$$\n", "\\begin{equation}\n", "\\frac{1}{2}v^2-g z = \\text{constant}\n", "\\end{equation}\n", "$$\n", "\n", "To get rid of that pesky constant, we can choose the reference horizontal line at the level that makes the constant energy equal to zero, so $v^2 = 2 g z$. That helps us re-write the phugoid equation in terms of $z$ as follows:\n", "\n", "$$\n", "\\begin{equation}\n", "\\frac{z}{z_t}-\\cos \\theta = \\frac{2z}{R}\n", "\\end{equation}\n", "$$\n", "\n", "Let $ds$ represent a small arc-length of the trajectory. We can write \n", "\n", "$$\n", "\\begin{equation}\n", "\\frac{1}{R} = \\frac{d\\theta}{ds} \\quad \\text{and}\\quad \\sin\\theta = -\\frac{dz}{ds}\n", "\\end{equation}\n", "$$\n", "\n", "Employing the chain rule of calculus,\n", "\n", "$$\n", "\\begin{equation}\n", "\\frac{1}{R} = \\frac{d\\theta}{ds} = \\frac{dz}{ds}\\frac{d\\theta}{dz} = -\\sin \\theta\\frac{d\\theta}{dz}\n", "\\end{equation}\n", "$$\n", "\n", "Multiply the phugoid equation by $\\frac{1}{2\\sqrt{z}}$ to get:\n", "\n", "$$\n", "\\begin{equation}\n", "\\frac{\\sqrt{z}}{2z_t} - \\frac{\\cos\\theta}{2\\sqrt{z}} = \\frac{\\sqrt{z}}{R}\n", "\\end{equation}\n", "$$\n", "\n", "Substituting for $1/R$ on the right hand side and bringing the cosine term over to the right, we get:\n", "\n", "$$\n", "\\begin{equation}\n", "\\frac{\\sqrt{z}}{2z_t} = \\frac{\\cos \\theta}{2 \\sqrt{z}} - \\sqrt{z} \\sin \\theta \\frac{d\\theta}{dz}\n", "\\end{equation}\n", "$$\n", "\n", "The right-hand-side is an exact derivative! We can rewrite it as:\n", "\n", "$$\n", "\\begin{equation}\n", "\\frac{d}{dz} \\left(\\sqrt{z}\\cos\\theta \\right) = \\frac{\\sqrt{z}}{2z_t}\n", "\\end{equation}\n", "$$\n", "\n", "Integrating this equation, we add an arbitrary constant, chosen as $C\\sqrt{z_t}$ which (after dividing through by $\\sqrt{z}$) gives:\n", "\n", "$$\n", "\\begin{equation}\n", "\\cos \\theta = \\frac{1}{3}\\frac{z}{z_t} + C\\sqrt{\\frac{z_t}{z}}\n", "\\end{equation}\n", "$$\n", "\n", "Taking the derivative of both sides of equation (19) and applying the relations from equation (15) yields:\n", "\n", "$$\n", "\\begin{equation}\n", "\\frac{z_t}{R} = \\frac{1}{3} - \\frac{C}{2}\\sqrt{\\frac{z_t^3}{z^3}}\n", "\\end{equation}\n", "$$\n", "\n", "Make sure you have followed the derivation, and perhaps write it out on paper!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Phugoid Curves" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Equation (15) is non-linear, which usually means we are hard-pressed to write a clean expression for the variable of interest, $z$. In fact, Lanchester himself said that he was unable to _\"reduce this expression to a form suitable for co-ordinate plotting.\"_ If the great polymath couldn't do it, we can't either!\n", "\n", "But Lanchester _was_ able to plot a suitable approximation of the phugoid flight path using what he called the \"trammel\" method. If you're interested in seeing how he did it, his explanation begins on page [48 of Aerodonetics](http://books.google.com/books?id=6hxDAAAAIAAJ&pg=PA49&lpg=PA48&dq=aerodonetics+the+use+of+the+trammel&source=bl&ots=lB6EVKYQuT&sig=aVE2kiDWZoWftaWczMIrcYftMOs&hl=en&sa=X&ei=gTD_U82fGYjzgwT3moGwCQ&ved=0CCAQ6AEwAA#v=onepage&q=aerodonetics%20the%20use%20of%20the%20trammel&f=false). It's a trip.\n", "\n", "Lanchester used Equations (15) and (16) to solve for the constant $C$ and the radius of curvature $R$ and then iteratively plotted small arcs of the phugoid path. By hand.\n", "\n", "We wrote a neat little code that duplicates the manual trammel method, but it might be a bit much for you to absorb in the first lesson. If you want to look it over, you are more than welcome to. If you are just starting with Python, skip it for the moment and we'll return to it at the end of this module. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Plotting the flight path" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As we mentioned, we wrote a Python code to reproduce programmatically what Lanchester did graphically. Here's a neat feature of Jupyter Notebooks: you can run external programs with the magical keyword ... wait for it ... `run`. And the jargon of Jupyter _is_ to call this \"magic.\" In fact, there are a bunch of [magic functions](https://ipython.readthedocs.io/en/stable/interactive/index.html) that you will learn about. They will make you a happy camper.\n", "The line `%matplotlib inline` tells Jupyter Notebook to show the plots inline.\n", "\n", "Let's do it:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%run phugoid.py\n", "%matplotlib inline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This code cell loaded our simulated-trammel code, `phugoid.py`. The code defined a function for you in the background, called `plot_flight_path`, taking three positional inputs: $z_t$, $z$ and $\\theta$. \n", "\n", "Look again at Equation (15), where we take the positive square root. There are several possibilities, depending on the value that the constant $C$ takes. \n", "\n", "* There are no physical solutions for $C>2/3$, because it would result in $\\cos\\theta>1$. \n", "\n", "* If $C=2/3$, then the solution is a horizontal straight line, because $\\cos\\theta=1$, $\\theta=0$ and $R=\\infty$.\n", "\n", "* Any value of $C$ for which $0 < C < \\frac{2}{3}$ will produce \"trochoidal\"-like paths. What does this look like? Let's use our custom function `plot_flight_path` to find out!" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Initial conditions: zt = 64.0, z0 = 16.0, theta0 = 0.0.\n", "plot_flight_path(64.0, 16.0, 0.0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Cool! Note that the plot title tells us what the calculated value of $C$ was for our input conditions. We have a value of $C$ between $0$ and $\\frac{2}{3}$ and our path is trochoidal, like we announced it would be.\n", "\n", "* For negative values of $C$, the resultant flight path consists of a series of loops. Let's try it out!" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Initial conditions: zt = 64.0, z0 = 16.0, theta0 = 180.0.\n", "plot_flight_path(64.0, 16.0, 180.0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can play around with the input values and see what kind of behavior results. Just note that any value of $C > \\frac{2}{3}$ will result in $\\cos \\theta > 1$, which doesn't exist. Python will probably throw a few errors if you hit that condition, but just try again!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* The last case is $C = 0$. Take another look at Equation (16) and plug in $C = 0$, what should happen? It looks like it will just reduce to \n", "\n", "$$R = 3z_t$$\n", "\n", "It's a constant radius of curvature! In fact, this solution is a series of semi-circles, with a cusp between them. One way to force $C = 0$ that we can figure out from Equation (15), is to make:\n", "\n", "\n", "$$z = 3z_t, \\quad \\theta = 0$$" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Initial conditions: zt = 16.0, z0 = 48.0, theta0 = 0.0.\n", "plot_flight_path(16.0, 48.0, 0.0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "That looks an awful lot like a quarter circle. And what's the radius of the arc? It's $r = 48 = 3z_t$.\n", "\n", "We can also get a semi-circle out of our simulated trammel by changing to another configuration where $C$ is (near) zero. Here's one example:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Initial conditions: zt = 64.0, z0 = 16.0, theta0 = -90.0.\n", "plot_flight_path(64.0, 16.0, -90.0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "That is so nice. We have reproduced the trajectories that Lanchester found more than a hundred years ago, painstakingly drawing them by hand with a contraption called a \"trammel.\" It must have taken him days!\n", "\n", "Here is how the different phugoid curves are drawn in von Kármán's book, _Aerodynamics_ (1957). He never says _how_ he drew them, but we're guessing by hand, also. We did pretty good!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "![Image](./figures/vonKarman-phugoids.png)\n", "\n", "#### Figure 4. Phugoid curves in von Kármán (1957)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the [next notebook](https://nbviewer.jupyter.org/github/numerical-mooc/numerical-mooc/blob/master/lessons/01_phugoid/01_02_Phugoid_Oscillation.ipynb) of this series, we'll look at the differential equation that arises when you consider small perturbations on the horizontal phugoid, and we'll learn to numerically integrate that to get the flight paths." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## References" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "1. Lanchester, F. W. _Aerodonetics_, D. van Nostrand Company: New York, 1909. On the public domain. [Get it from Google Books](http://books.google.com/books?id=6hxDAAAAIAAJ&pg=PP1#v=onepage&q&f=false).\n", "\n", "2. Milne-Thompson, L. M. _Theoretical Aerodynamics_, Dover 2012 reprint of the revised 1966 edition. [Read on Google Books](http://books.google.com/books?id=EMfCAgAAQBAJ&lpg=PP1&pg=PP1#v=onepage&q&f=false) (see section 18.5)\n", "\n", "3. Sinha, N. K. and Ananthkrishnan, N. _Elementary Flight Dynamics with an introduction to Bifurcation and Continuation Methods_, CRC Press, 2013. [Read on Google Books](http://books.google.com/books?id=yXL6AQAAQBAJ&lpg=PP1&pg=PP1#v=onepage&q&f=false) (see chapter 5)\n", "\n", "4. von Kármán, T. _Aerodynamics_, Dover 2004 reprint of the 1957 2nd edition. (see pages 149–151)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "---\n", "\n", "###### The cell below loads the style of this notebook. " ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Execute this cell to load the notebook's style sheet, then ignore it.\n", "from IPython.core.display import HTML\n", "css_file = '../../styles/numericalmoocstyle.css'\n", "HTML(open(css_file, 'r').read())" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (MOOC)", "language": "python", "name": "py36-mooc" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.6" } }, "nbformat": 4, "nbformat_minor": 1 }