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 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)" Luke Nashs 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 •