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Aircraft Flight Dynamics
Robert Stengel, Princeton University, 2012" Copyright 2012 by Robert Stengel. All rights reserved. For educational use only.! http://www.princeton.edu/~stengel/MAE331.html! ! Dynamics & Control of Atmospheric Flight ! Configuration Aerodynamics ! Aircraft Performance ! Flight Testing and Flying Qualities ! Aviation History Details • Lecture: 3-4:20, D-221, Tue & Thu, E-Quad • Precept (as announced): 7-8:20, D-221, Mon • Engineering, science, & math • Case studies, historical context • ~6 homework assignments • Office hours: 1:30-2:30, MW, D-202, or any time the door is open • Assistants in Instruction: Carla Bahri, Paola Libraro: Office hours: TBD • GRADING – Assignments: 30% – First-Half Exam: 15% – Second-Half Exam: 15% – Te r m Pap e r: 30 % – Class participation: 10% – Quick Quiz (5 min): ?% • Lecture slides – pdfs from all 2010 lectures are available now at http://www.princeton.edu/~stengel/MAE331.html – pdf for current (2012) lecture will be available on Blackboard after the class Syllabus, First Half ! Introduction, Math Preliminaries ! Point Mass Dynamics ! Aviation History ! Aerodynamics of Airplane Configurations ! Cruising Flight Performance ! Gliding, Climbing, and Turning Performance ! Nonlinear, 6-DOF Equations of Motion ! Linearized Equations of Motion ! Longitudinal Dynamics ! Lateral-Directional Dynamics Details, reading, homework assignments, and references at http://blackboard.princeton.edu/" Syllabus, Second Half ! Analysis of Linear Systems ! Time Response ! Root Locus Analysis of Parameter Variations ! Transfer Functions and Frequency Response ! Aircraft Control and Systems ! Flight Testing ! Advanced Problems in Longitudinal Dynamics ! Advanced Problems in Lateral-Directional Dynamics ! Flying Qualities Criteria ! Maneuvering and Aeroelasticity ! Problems of High Speed and Altitude ! Atmospheric Hazards to Flight Text and References • Principal textbook: – Flight Dynamics, RFS, Princeton University Press, 2004 – Used throughout • Supplemental references – Airplane Stability and Control, Abzug and Larrabee, Cambridge University Press, 2002 – Virtual textbook, 2012 Stability and Control Case Studies" Ercoupe" Electra" F-100" Flight Tests Using Balsa Glider and Cockpit Flight Simulator • 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, … • Compare actual flight of the glider with trajectory simulation Assignment #1 due: Friday, September 21 • 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. ! Luke Nashs Biplane Glider Flight #1 (MAE 331, 2008)" • Can determine height, range, velocity, flight path angle, and pitch angle from sequence of digital photos (QuickTime)" Luke Nashs Biplane Glider Flight #1 (MAE 331, 2008)" Electronic Devices in Class • Silence all cellphones and computer alarms • If you must make a call or send a message, you may leave the room to do so • No checking or sending text, tweets, etc. – No social networking – No surfing • Pencil and paper for note-taking • American Institute of Aeronautics and Astronautics! – largest aerospace technical society! – 35,000 members! • https://www.aiaa.org! • Benefits of student membership ($20/yr)! – Aerospace America magazine! – Daily Launch newsletter! – Monthly Members Newsletter, Quarterly Student Newsletter! – Aerospace Career Handbook! – Scholarships, design competitions, student conferences! MAE department will reimburse dues when you join! i.e., it’s free!" Goals for Design" • Shape of the airplane determined by its purpose" • Handling, performance, functioning, and comfort" • Agility vs. sedateness" • Control surfaces adequate to produce needed moments" • 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" Inhabited Air Vehicles" Uninhabited Air Vehicles (UAV)" Quick Quiz #1 First 5 Minutes of Next Class ! Briefly describe the differences between one of the following groups of airplanes: A. Boeing B-17 vs. Northrop YB-49 vs. North American B-1 B. Piper Cub vs. Beechcraft Bonanza vs. Cirrus SR20 C. Douglas DC-3 vs. Boeing 707 vs. Airbus A320 D. Lockheed P-38 vs. North American F-86 vs. Lockheed F-35 ! Use Wikipedia to learn about all of these planes ! Group (A or B or C or D) will be chosen by coin flip in next class ! Be sure to bring a pencil and paper to class Introduction to Flight Dynamics Airplane Components " Airplane Rotational Degrees of Freedom" Airplane Translational Degrees of Freedom" Axial Velocity" Side Velocity" Normal " Velocity" Phases of Flight" Flight of a Paper Airplane 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" • Equations of motion integrated numerically to estimate the flight path" 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" Assignment #2 • Compute the trajectory of a balsa glider Gliding Flight" Configuration Aerodynamics" Math Preliminaries Notation for Scalars and Vectors " • Scalar: usually lower case: a, b, c, …, x, y, z " a = 2 −7 16 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ; x = x 1 x 2 x 3 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ; y = a b c d ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ • Vector: usually bold or with underbar: x or x" • Ordered set" • Column of scalars" • Dimension = n x 1" a = 12; b = 7; c = a + b = 19; x = a + b 2 = 12 + 49 = 61 Matrices and Transpose" x = p q r ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ; A = a b c d e f g h k l m n ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ A T = a d g l b e h m c f k n ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ x T = x 1 x 2 x 3 ⎡ ⎣ ⎤ ⎦ • Matrix: usually bold capital or capital: F or F" • Dimension = (m x n)" • Transpose: interchange rows and columns" 3 × 1 ( ) 4 × 3 ( ) Multiplication " ax T = ax 1 ax 2 ax 3 ⎡ ⎣ ⎤ ⎦ ax = xa = ax 1 ax 2 ax 3 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ • Operands must be conformable" • Multiplication of vector by scalar is associative, commutative, and distributive" • Could we add ?" x + a ( ) • Only if" dim x ( ) = 1 × 1 ( ) a x + y ( ) = x + y ( ) a = ax + ay ( ) dim x ( ) = dim y ( ) Addition " x = a b ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ; z = c d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ • Conformable vectors and matrices are added term by term " x + z = a + c b + d ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Inner Product " x T x = x • x = x 1 x 2 x 3 ⎡ ⎣ ⎤ ⎦ x 1 x 2 x 3 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ • Inner (dot) product of vectors produces a scalar result" (1 × m)(m × 1) = (1 × 1) = (x 1 2 + x 2 2 + x 3 2 ) • Length (or magnitude) of vector is square root of dot product" = (x 1 2 + x 2 2 + x 3 2 ) 1/2 Vector Transformation " y = Ax = 2 4 6 3 −5 7 4 1 8 −9 −6 −3 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ x 1 x 2 x 3 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ (n × 1) = (n × m)(m × 1) • Matrix-vector product transforms one vector into another " • Matrix-matrix product produces a new matrix" = 2x 1 + 4x 2 + 6x 3 ( ) 3x 1 − 5x 2 + 7x 3 ( ) 4 x 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 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ Matrix Inverse" x y z ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 2 = cos θ 0 − sin θ 0 1 0 sin θ 0 cos θ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ x y z ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 1 Transformation" Inverse Transformation" x y z ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 1 = cos θ 0 sin θ 0 1 0 −sin θ 0 cos θ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ x y z ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 2 x 2 = Ax 1 x 1 = A −1 x 2 Matrix Identity and Inverse" I 3 = 1 0 0 0 1 0 0 0 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ AA −1 = A −1 A = I y = Iy • Identity matrix: no change when it multiplies a conformable vector or matrix" • A non-singular square matrix multiplied by its inverse forms an identity matrix" AA −1 = cos θ 0 −sin θ 0 1 0 sin θ 0 cos θ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ cos θ 0 −sin θ 0 1 0 sin θ 0 cos θ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ −1 = cos θ 0 −sin θ 0 1 0 sin θ 0 cos θ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ cos θ 0 sin θ 0 1 0 −sin θ 0 cos θ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = 1 0 0 0 1 0 0 0 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ Dynamic Systems" 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 vectors" Sensors! Actuators! Mathematical Models of Dynamic Systems are Differential Equations" x(t ) dx(t ) dt = f[x(t ),u(t ),w(t ),p(t ),t ] y(t) = h[x(t),u(t)] z(t ) = y(t ) + n(t) Continuous-time dynamic process: Vector Ordinary Differential Equation" Output Transformation" Measurement with Error" dim x ( ) = n × 1 ( ) dim f ( ) = n × 1 ( ) dim u ( ) = m × 1 ( ) dim w ( ) = s × 1 ( ) dim p ( ) = l × 1 ( ) dim y ( ) = r ×1 ( ) dim h ( ) = r ×1 ( ) dim z ( ) = r ×1 ( ) dim n ( ) = r ×1 ( ) [...]... business jet aircraft (MATLAB)" – http://www.princeton.edu/ ~stengel/ FlightDynamics.html" – http://www.princeton.edu/ ~stengel/ FDcodeB.html" • Linear system analysis (MATLAB)" • Paper airplane simulation (MATLAB)" • Helpful Resources " • Web pages" – http://blackboard.princeton.edu/" – http://www.princeton.edu/ ~stengel/ MAE331.html" – http://www.princeton.edu/ ~stengel/ FlightDynamics.html" • Princeton University... a business jet aircraft (Excel)" – http://www.princeton.edu/ ~stengel/ FDcodeC.html" – http://www.princeton.edu/ ~stengel/ PaperPlane.html" – http://lib-terminal.princeton.edu/ejournals/by_title_zd.asp" • NACA/NASA and AIAA pubs" – http://ntrs.nasa.gov/search.jsp" – http://www.princeton.edu/ ~stengel/ Example261.xls" Primary Learning Objectives More Learning Objectives " ! Introduction to the performance,... Airspeed! Flight Path Angle! Pitch Rate! Angle of Attack! • Note change in time scale" Airspeed! Pitch Rate! Simplified Lateral Modes of Motion " Flight Path Angle! Angle of Attack! Simplified Lateral Modes of Motion " Dutch-Roll Mode" Yaw Rate! Roll and Spiral Modes" Sideslip Angle! Roll Rate! Roll Angle! Flight Dynamics Book and Computer Code " • All programs are accessible from the Flight Dynamics. .. fixed-wing aircraft ranging from micro-uninhabited air vehicles through general aviation, jet transport, and fighter aircraft to re-entry vehicles ! Detailed evaluation of the linear and nonlinear flight characteristics of a specific aircraft type." ! Improved skills for presenting ideas, orally and on paper." ! Improved ability to analyze complex, integrated problems." ! Understanding of aircraft equations...Next Time: Point-Mass Dynamics and Aerodynamic/Thrust Forces Supplemental Material Reading: Flight Dynamics for Lecture 1: 1-27 for Lecture 2: 29-34, 38-53, 59-65, 103-107 Virtual Textbook, Parts 1 and 2 Examples of Airplane Dynamic System Models " Ordinary Differential... complex, integrated problems." ! Understanding of aircraft equations of motion, configuration aerodynamics, and methods for analysis of linear and nonlinear systems ! Demonstrated computing skills, through thorough knowledge and application of MATLAB." ! Facility in evaluating aircraft kinematics and dynamics, flight envelopes, trim conditions, maximum range, climbing/diving/turning flight, inertial... longitudinal and lateral-directional transients, transfer functions, state-space models, and frequency response." ! Appreciation of the historical context within which past aircraft have been designed and operated, providing a sound footing for the development of future aircraft . Wikipedia to learn about all of these planes ! Group (A or B or C or D) will be chosen by coin flip in next class ! Be sure to bring a pencil and paper to class Introduction to Flight Dynamics Airplane. of Motion" Flight Dynamics Book and Computer Code" • All programs are accessible from the Flight Dynamics web page" – http://www.princeton.edu/~stengel/FlightDynamics.html" • . only.! http://www.princeton.edu/~stengel/MAE331.html! ! Dynamics & Control of Atmospheric Flight ! Configuration Aerodynamics ! Aircraft Performance ! Flight Testing and Flying Qualities ! Aviation History Details •
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