# Flight Simulation

Lecture: Wednesday, 17.00-19.00 (Sal 139),
Exercise: Wednesday, 19.00-21.00 (Sal 107)
First lecture is on 08.10.2014. Students are expected to attend the lectures and the exercises.

## Prerequisites

You must know basics of differential calculus, linear algebra, mechanics, and computer graphics. You must be fluent in C++. This is a serious course, which involves a lot of mathematics!

## Topics

First five lectures: Differential equations. Numerical methods for solving differential equations. Order of a numerical method. Runge-Kutta Methods. We apply RK methods to orbits, Lagrange points, and rocket trajectories. Analytic solutions of systems of linear differential equations. Qualitative behavior of solutions.

Next two lectures: Basics of mechanics. The notions of mass center, torque (moment), and momentum. Definition of rigid body. Derivation of laws for rigid body movement.

Two lectures: Properties of airfoils. Typical lift/drag curves. Notion of aerodynamic center, and its importance for stability. Theory of longitudinal stability. Kutta-Joukowski law: Lift = Uniform Flow X Circulation.

One/two lectures: Application of rigid-body laws to airplanes. Theory of stability in three dimensions. The four stability modes. How to enhance stability with automatic controls. (Autopilots.)

One lecture: Representation of orientation by quaternions. Role of quaternions in simulation and computer graphics.

One lecture: Detecting contacts with the ground or other objects. There exist collisions and there exist controlled ground contacts. In order to be able to simulate landings and take offs, one need to detect when the airplane touches the ground, in which way it touches the ground, and what forces these contacts create.

One lecture: Modelling of wheels.

## SFML (Simple and Fast Multimedia Library)

I use SFML as Window Manager, and as interface to OpenGL. SMFL can be obtained from here . You need version 2.0 or higher. Installing SFML in Ubuntu is easy, because it has the correct version in the package manager. Unfortunately, Debian still has version 1.6. The differences are not big, but they are irritating, so try to get version 2.0 or later.

## Open GL

SFML supports computer graphics through OpenGL. The homepage of Andrzej Lukaszewski contains a lot of pointers to openGL. The Red Book used to be the main source for learning OpenGL. It is outdated, because the standard commands are deprecated, and one should use shading language. Even when the commands in the Red Book are outdated, the algorithms are still valid, so it is still useful to look at the first 5 sections.

## Lectures

• 8.10.2014. General introduction . Movement of a point mass under gravity. Movement of a group of point masses under each other's gravity. Modelling of rockets. Ciolkowski rocket equation. slides .
• 15.10.2014. The Oberth-Effect. (Using a rocket engine at low potential energy is more effective than using one at high potential energy). Differential equation for suspension bridge, and differential equation for free hanging chain. Reducing the order of a differential equation, making it autonomous, and Euler's method for numerical solving.
• 22.10.2014. How to check invariants of differential equations. How to check preservation properties. Runge-Kutta Methods for solving differential equations. (Slides). Demonstration of the order of a numerical method on the equation for the catenoid. chain.cpp and runge_kutta.h.
• 29.10.2014. I wanted to do some rocket and planet simulations using RK-methods, but I messed up the planning, so this topic has to be postponed until next week. This week I will therefore treat quaternions.
• 05.11.2014. More about quaternions. I also use the occasion to show some videos of historic events related to space flight:
• 12.11.2014. Spaceflight: How to simulate rockets (Saturn V). I show that with simple simulation methods, it is possible to get numbers that are very close to the published numbers. Also, one can observe many effects that exist in real.

Second part: Development of two dimensional aircraft model: Basic properties of airfoils. Circulation theory. Standard formulas for lift and drag.

• 19.11.2014. More about airfoils: center of pressure, moment coefficient, aerodynamic center. Inherent instability of airfoils. Mechanics of rigid objects in two dimensions. Slides.
• 26.11.2014. I show you code that implements the two dimensional flight model. I demonstrate the two stability modes that expressible in 2D. (The phugoid and the short oscillation.)
• On 10.12.2014, there will be an excursion to an Airbus A320 Simulator in Dresden. Please reserve this day, because traveling to Dresden will take the whole day. There is room for 20 students. Taking part results in 20 credit points (the equivalent of two exercises.)
• 17.12.2014. Final remarks about two dimensional flight model. Development of Mechanics of rigid objects in 3D. Behaviour of sunset/sunrise times near the winter solstice.
• 14.01.2015. Development of Mechanics of rigid objects in 3D. (I made some additions to the text mechanics.pdf.)
• 21.01.2015. More about mechanics of rigid objects. Flights simulation in three dimensions. Slides.
• 28.01.2015. More about flight simulation in three dimensions. Collision detection. Slides.
• 04.02.2015. Last lecture. Topic: wheels must roll for victory! Slides.
• On 11.02.2015, there will still lab in room 107, from 17.00 to 19.00.

## Exercises

• Most of exercise 0 was made in class.
• Exercise list 1. This exercise must be handed in on 22.10.2014, and WILL BE GRADED. Make it on separate paper.
• Exercise list 2 (corrected version). This exercise must be handed in on 29.10.2014. Part 2c and 3 can be handed in on 05.11.2014.
• Exercise list 3, which must be handed in on 12.11.2014.
• Exercise list 4, which must be handed in on 19.11.2014. (You need this drawing.)
• Exercise list 5, which must be handed in on 03.12.2014.
• Exercise list 6, which must be handed in on 17.12.2014.
• Exercise list 7, which must be shown on 21.01.2015 in Lab 107. You need this code.
• Exercise list 8. You need this two dimensional simulation. This is the last exercise. I will publish the point system very soon.

pictures.

## Literature

• Mechanics of Flight, A.C. Kermode, 11th Edition, (Revised and Edited by R.H.Barnard and D.R.Philpott) Pearson/Prentice Hall, UK, 2006.
• Aircraft Control and Simulation, Brian Stevens and Frank Lewis, John Wiley and Sons Inc., 1992.
• Fluid Mechanics DeMYSTiFieD (A Self-Teaching Guide), Merle C. Potter, McGraw Hill, 2009.
• Numerical Methods for Ordinary Differential Equations, J.C. Butcher, Wiley and Sons, 2003.
• Mechanics of Flight (Second Edition), Warren F. Philips, John Wiley and Sons, Inc., 2010.
• Analyse, J.H.J. Almering e.a., VSSD, Delftse Uitgevers Maatschappij, the Netherlands, 1984.