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% �–  Term Paper: 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 =2−716

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥; x =

x1x2x3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥; y =

abcd

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

•  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 + b2 = 12 + 49 = 61

Matrices and Transpose"

x =pqr

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥; A =

a b cd e fg h kl m n

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

AT =a d g lb e h mc f k n

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xT = x1 x2 x3⎡⎣

⎤⎦

•  Matrix: usually bold capital or capital: F or F"•  Dimension = (m x n)"

•  Transpose: interchange rows and columns"3×1( ) 4 × 3( )

Multiplication "

axT = ax1 ax2 ax3⎡⎣

⎤⎦

ax = xa =ax1ax2ax3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

•  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 = ab

⎡

⎣⎢

⎤

⎦⎥ ; z = c

d⎡

⎣⎢

⎤

⎦⎥

•  Conformable vectors and matrices are added term by term "

x + z = a + cb + d

⎡

⎣⎢

⎤

⎦⎥

Inner Product "

xTx = x • x = x1 x2 x3⎡⎣

⎤⎦

x1x2x3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

•  Inner (dot) product of vectors produces a scalar result"

(1× m)(m ×1) = (1×1)

= (x12 + x2

2 + x32 )

•  Length (or magnitude) of vector is square root of dot product"

= (x12 + x2

2 + x32 )1/2

Vector Transformation "

y = Ax =

2 4 63 −5 74 1 8−9 −6 −3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

x1x2x3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

(n ×1) = (n × m)(m ×1)

•  Matrix-vector product transforms one vector into another "•  Matrix-matrix product produces a new matrix"

=

2x1 + 4x2 + 6x3( )3x1 − 5x2 + 7x3( )4x1 + x2 + 8x3( )

−9x1 − 6x2 − 3x3( )

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

=

y1y2y3y4

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

Derivatives and Integrals of Vectors"

•  Derivatives and integrals of vectors are vectors of derivatives and integrals"

dxdt

=

dx1dt

dx2dt

dx3dt

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

x∫ dt =

x1∫ dt

x2∫ dt

x3∫ dt

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

Matrix Inverse"

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥2

=cosθ 0 − sinθ0 1 0sinθ 0 cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥1

Transformation"

Inverse Transformation"

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥1

=cosθ 0 sinθ0 1 0

− sinθ 0 cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥2

x2 = Ax1

x1 = A−1x2

Matrix Identity and Inverse"

I3 =1 0 00 1 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

AA−1 = A−1A = 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 =cosθ 0 − sinθ0 1 0sinθ 0 cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

cosθ 0 − sinθ0 1 0sinθ 0 cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−1

=cosθ 0 − sinθ0 1 0sinθ 0 cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

cosθ 0 sinθ0 1 0

− sinθ 0 cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=1 0 00 1 00 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( )

Next Time: �Point-Mass Dynamics and Aerodynamic/Thrust Forces �

�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 �

Supplemental Material

Ordinary Differential Equations"

dx(t)dt

= f x(t),u(t),w(t)[ ]

dx(t)dt

= f x(t),u(t),w(t),p(t),t[ ] dx(t)dt

= F(t)x(t) +G(t)u(t) + L(t)w(t)

dx(t)dt

= Fx(t) +Gu(t) + Lw(t)

Examples of Airplane Dynamic System Models"

•  Nonlinear, Time-Varying"–  Large amplitude motions"–  Significant change in mass"

•  Nonlinear, Time-Invariant"–  Large amplitude motions"–  Negligible change in mass"

•  Linear, Time-Varying"–  Small amplitude motions"–  Perturbations from a dynamic

flight path"

•  Linear, Time-Invariant"–  Small amplitude motions"–  Perturbations from an

equilibrium flight path"

Simplified Longitudinal Modes of Motion"Phugoid (Long-Period) Mode"

Airspeed! Flight Path Angle!

Pitch Rate! Angle of Attack!

Short-Period Mode"

Airspeed! Flight Path Angle!

Pitch Rate! Angle of Attack!

•  Note change in time scale"

Simplified Longitudinal Modes of Motion"

Simplified Lateral Modes of Motion"Dutch-Roll Mode"

Yaw Rate! Sideslip Angle!

Roll and Spiral Modes"

Roll Rate! Roll Angle!

Simplified Lateral Modes 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"•  ... or directly"•  Errata for the book are listed there"•  6-degree-of-freedom nonlinear simulation of a business jet

aircraft (MATLAB)"–  http://www.princeton.edu/~stengel/FDcodeB.html"

•  Linear system analysis (MATLAB)"–  http://www.princeton.edu/~stengel/FDcodeC.html"

•  Paper airplane simulation (MATLAB)"–  http://www.princeton.edu/~stengel/PaperPlane.html"

•  Performance analysis of a business jet aircraft (Excel)"–  http://www.princeton.edu/~stengel/Example261.xls"

Helpful Resources"

•  Web pages"– http://blackboard.princeton.edu/"– http://www.princeton.edu/~stengel/MAE331.html"– http://www.princeton.edu/~stengel/FlightDynamics.html"

•  Princeton University Engineering Library (paper and on-line)"– http://lib-terminal.princeton.edu/ejournals/by_title_zd.asp"

•  NACA/NASA and AIAA pubs"– http://ntrs.nasa.gov/search.jsp"

Primary Learning Objectives�!  Introduction to the performance, stability, and control of �fixed-wing� aircraft ranging from micro-uninhabited air vehicles through general aviation, jet transport, and fighter aircraft to re-entry vehicles.�

!  Understanding of aircraft equations of motion, configuration aerodynamics, and methods for analysis of linear and nonlinear systems.�

!  Appreciation of the historical context within which past aircraft have been designed and operated, providing a sound footing for the development of future aircraft.�

More Learning Objectives"!  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."

!  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 properties, stability-and-control derivatives, longitudinal and lateral-directional transients, transfer functions, state-space models, and frequency response."