# Aircraft Flight Dynamics!

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1 Aircraft Flight Dynamics Robert Stengel, Princeton University, 2014 Course Overview Introduction to Flight Dynamics Math Preliminaries Copyright 2014 by Robert Stengel. All rights reserved. For educational use only. At the End of the Course, you should be able to: Understand aircraft configuration aerodynamics, performance, stability, and control Estimate an aircraft s aerodynamic characteristics from geometric and inertial properties Analyze linear and nonlinear dynamic systems Recognize airplane modes of motion and their significance Compute aircraft motions Appreciate historical development of aviation

2 Syllabus, First Half Introduction, Math Preliminaries Point Mass Dynamics Aerodynamics of Airplane Configurations Cruising Flight Performance Gliding, Climbing, and Turning Performance Wind Tunnel and Flight Testing Nonlinear, 6-DOF Equations of Motion Aircraft Control Devices and Systems Linearized Equations of Motion Longitudinal Dynamics Lateral-Directional Dynamics Details, reading, homework assignments, and references at Syllabus, Second Half Analysis of Linear Systems Time Response Transfer Functions and Frequency Response Root Locus Analysis Advanced Problems in Longitudinal Dynamics Advanced Problems in Lateral-Directional Dynamics Flying Qualities Criteria Maneuvering at High Angles and Rates Aeroelasticity and Fuel Slosh Problems of High Speed and Altitude Atmospheric Hazards to Flight

3 Your interested in MAE 331 because? Text and References Science, Engineering, and Math: Flight Dynamics, RFS, Princeton University Press, html Case Studies, Historical Context Airplane Stability and Control, Abzug and Larrabee, Cambridge University Press, 2002 Supplemental reference Virtual reference book, 2013

4 Quick Quizzes First 5 Minutes of 10 Classes One question about the lectures and reading assignments from the previous week Largely qualitative but may require simple calculations Be sure to bring a pencil, paper, and calculator to class Homework Assignments All assignments will be done in groups of 2 or 3 students Team members for each assignment will be different chosen using a spreadsheet and random number generator (TBD) Each member of each team will receive the same grade as the others

5 Flight Tests Using Balsa Glider and Cockpit Flight Simulator Compare actual flight of the glider with trajectory simulation Flight envelope of full-scale aircraft simulation Maximum speed, altitude ceiling, stall speed, Performance Time to climb, minimum sink rate, Turning Characteristics Maximum turn rate, Assignment 1 due: September 19, 2014 Document the physical characteristics and flight behavior of a balsa glider Everything that you know about the physical characteristics of the glider Everything that you know about the flight characteristics of the glider 2-person team, joint write-up

6 Luke Nash s Biplane Glider Flight 1 (MAE 331, 2008) Luke Nash s Biplane Glider Flight 1 (MAE 331, 2008) Can determine height, range, velocity, flight path angle, and pitch angle from sequence of digital photos (QuickTime)

7 Stability and Control Case Studies F-100 Ercoupe Electra Reading Assignments All students are expected to do all reading assignments before class Reading for Case Studies and Historical Context, Airplane Stability and Control 10-minute synopses by groups of 3 students Principal subject and scope of the chapter Technical ideas needed to understand the chapter When did the events covered in the chapter occur? Three main takeaway points or conclusions Three most surprising or remarkable facts 1 st synopsis: Sept 23 rd, team members TBD

8 Goals for Airplane Design Shape of the airplane determined by its purpose Safety, handling, performance, functioning, and comfort Agility vs. sedateness Control surfaces adequate to produce needed moments (i.e., torques) Center of mass location too far forward increases unpowered control-stick forces too far aft degrades static stability Configuration Driven By The Mission and Flight Envelope

9 Inhabited Air Vehicles Uninhabited Air Vehicles (UAV)

10 Introduction to Flight Dynamics Airplane Components

11 Airplane Rotational Degrees of Freedom Airplane Translational Degrees of Freedom Side Velocity Axial Velocity Normal Velocity

12 Flight of a Paper Airplane Phases of Flight

13 Flight of a Paper Airplane Example 1.3-1, Flight Dynamics Equations of motion integrated numerically to estimate the flight path Red: Equilibrium flight path Black: Initial flight path angle = 0 Blue: plus increased initial airspeed Green: loop Flight of a Paper Airplane Example 1.3-1, Flight Dynamics Red: Equilibrium flight path Black: Initial flight path angle = 0 Blue: plus increased initial airspeed Green: loop

14 Assignment 2 Compute the trajectory of a balsa glider Computer simulation of the equations of motion Compare to the actual flight of the glider (Assignment 1) Similar to the flight of a paper airplane 2-person team assignment Gliding Flight

15 Configuration Aerodynamics Math Preliminaries

16 Dynamic Systems Actuators Sensors Dynamic Process: Current state depends on prior state x = dynamic state u = input w = exogenous disturbance p = parameter t or k = time or event index Observation Process: Measurement may contain error or be incomplete y = output (error-free) z = measurement n = measurement error All of these quantities are multidimensional They can be expressed as vectors Notation for Scalars and Vectors Scalar: usually lower case: a, b, c,, x, y, z a = 12; b = 7; c = a + b = 19; x = a + b 2 = = 61 Vector: usually bold or with underbar: x or x Ordered set Column of scalars Dimension = n x 1 a = ; x = x 1 x 2 x 3 ; y = a b c d

17 Matrices and Transpose Matrix: usually bold capital or capital: F or F Dimension = (m x n) p x = q r a b c d e f ; A = g h k l m n ( 3 1) ( 4 3) Transpose: interchange rows and columns x T = x 1 x 2 x 3 A T = a d g l b e h m c f k n Multiplication Operands must be conformable Multiplication of vector by scalar is associative, commutative, and distributive ax = xa = ax 1 ax 2 ax 3 a( x + y) = ( x + y)a = ( ax + ay) dim( x) = dim( y) ax T = ax 1 ax 2 ax 3 Could we add ( x + a)? Only if dim( x) = ( 1 1)

18 Addition Conformable vectors and matrices are added term by term x = a b ; z = c d x + z = a + c b + d Inner Product Inner (dot) product of vectors produces a scalar result x T x = x x = x 1 x 2 x 3 (1 m)(m 1) = (1 1) x 1 x 2 x 3 = (x x x 3 2 ) Length (or magnitude) of vector is square root of dot product = (x x x 3 2 ) 1/2

19 Vector Transformation Matrix-vector product transforms one vector into another Matrix-matrix product produces a new matrix y = Ax = x 1 x 2 x 3 (n 1) = (n m)(m 1) = ( 2x 1 + 4x 2 + 6x 3 ) ( 3x 1 5x 2 + 7x 3 ) ( 4x 1 + x 2 + 8x 3 ) ( ) 9x 1 6x 2 3x 3 = y 1 y 2 y 3 y 4 Derivatives and Integrals of Vectors Derivatives and integrals of vectors are vectors of derivatives and integrals dx dt = dx 1 dt dx 2 dt dx 3 dt x dt = x 1 dt x 2 dt x 3 dt

20 Particle Matrix Inverse Transformation x 2 = Ax 1 x y z 2 = cos 0 (sin sin 0 cos x y z 1 Inverse Transformation x 1 = A 1 x 2 x y = z 1 cos 0 sin (sin 0 cos x y z 2 Matrix Identity and Inverse Identity matrix: no change when it multiplies a conformable vector or matrix I 3 = y = Iy A non-singular square matrix multiplied by its inverse forms an identity matrix AA 1 = A 1 A = I AA 1 = = cos 0 sin sin 0 cos cos 0 sin sin 0 cos = ( ( ( ( ( ( cos 0 sin sin 0 cos ( ( ( cos 0 sin sin 0 cos ( ( ( 1 ( ( (

21 Mathematical Models of Dynamic Systems are Differential Equations Continuous-time dynamic process: Vector Ordinary Differential Equation x(t) dx(t) dt = f[x(t),u(t),w(t),p(t),t] Output Transformation y(t) = h[x(t),u(t)] Measurement with Error z(t) = y(t) + n(t) dim x dim f dim u dim w dim p ( ) = ( n 1) ( ) = ( n 1) ( ) = ( m 1) ( ) = ( s 1) ( ) = ( l 1) dim y dim h dim z dim n ( ) = ( r 1) ( ) = ( r 1) ( ) = ( r 1) ( ) = ( r 1) Next Time: Learning Objectives Point-Mass Dynamics Aerodynamic/Thrust Forces Reading: Flight Dynamics Introduction, 1-27 The Earth s Atmosphere, Kinematic Equations, Forces and Moments, Introduction to Thrust,

22 Supplemental Material Flight Dynamics Book and Computer Code All programs are accessible from the Flight Dynamics web page or directly Errata for the book are listed there 6-degree-of-freedom nonlinear simulation of a business jet aircraft (MATLAB) Linear system analysis (MATLAB) Paper airplane simulation (MATLAB) Performance analysis of a business jet aircraft (Excel)

23 Helpful Resources Web pages Princeton University Engineering Library (paper and online) NACA/NASA pubs Course Learning Objectives (Accreditation Board for Engineering and Technology)

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