Details
Aircraft Flight Dynamics Robert Stengel, Princeton University, 2012
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 a ny time the door is open Assistants in Instruction: Carla Bahri, Paola Libraro:: Office hours: TBD Libraro
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Dynamics & Control of Atmospheric Flight Configuration Configura tion Aerodynamics Aircraft Performance Flight Testing and Flying Qualities Aviation History
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GRADING – – – – –
Copyright 2012 by Robert Stengel. All rights reserved. For education al use only. http://www.princeton.edu/~stengel/MAE331.html
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Introduction, Math Preliminaries Point Mass Dynamics Aviation History Aerodynamics of Airplane Configurations Configurations Cruising Flight Performance Gliding, Climbing, and an d Turning Turning Performance Nonlinear, 6-DOF Equations of Motion Linearized Equations of Motion Longitudinal Dynamics Lateral-Directional Dynamics
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Lecture slides –
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pdfs from all 20 10 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
Analysis of Linear Systems ! ! !
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Details, reading, homework assignments, and references at http://blackboard.princeton.edu/
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Syllabus, Second Half
Syllabus, First Half !
Assignments: 30% First-Half Exam: 15% Second-Half Exam: 15% Term Paper: 30% Class participation: 10% Quick Quiz (5 min): ?%
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Time Response Root Locus Analysis of Parameter Variations Transfer Tr ansfer Functions an d Frequency Response
Aircraft Control and Systems Flight Testing Advanced Advanc ed Probl Problems ems in Longitudinal Dynamics Advanced Problems in Lateral-Directional Dynamics Flying Qualities Criteria Maneuvering and Aeroelasticity Problems of High Speed and Altitude Atmospheric Hazards to Fligh Flightt
Text and References
Stability and Control Case Studies
Principal textbook:
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F-100
Flight Dynamics , RFS, Princeton University Press, 2004 Used throughout
Supplemental references
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Ercoupe
Airplane Stability and Control , Abzug and Larrabee, Cambridge University Press, 2002 Virtual textbook , 2012
Flight Tests Using Balsa Glider and Cockpit Flight Simulator
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Compare actual flight of the glider with trajectory simulation Flight envelope of full-scale aircraft simulation –
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Performance –
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Maximum speed, altitude ceiling, stall speed, … Time to climb, minimum sink rate, …
Turning Characteristics –
Maximum turn rate, …
Electra
Assignment #1
due: Friday, September 21
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Document the physical characteristics and flight behavior of a balsa glider. –
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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 )
Electronic Devices in Class • •
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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. – –
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Luke Nash s Biplane Glider Flight #1 (MAE 331, 2008)
No social networking No surfing
Pencil and paper for note-taking
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American Institute of Aeronautics and Astronautics –
largest aerospace technical society
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35,000 members
https://www.aiaa.org Benefits of student membership ($20 /yr) –
Aerospace America magazine
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Daily Launch newsletter
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Monthly Members Newsletter, Quarterly Student Newsletter
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Aerospace Career Handbook
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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
Configuration Driven By The Mission and Flight Envelope
– too far forward increases unpowered control-stick forces – too far aft degrades static stability
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
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Introduction to Flight Dynamics
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 an d paper to class
Airplane Components
Airplane Rotational Degrees of Freedom
Airplane Translational Degrees of Freedom
Phases of Flight
Side Velocity
Normal Velocity Axial Velocity
Flight of a Paper Airplane
Flight of a Paper Airplane Example 1.3-1, Flight Dynamics •
Equations of motion integrated numerically to estimate the flight path
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Red: Equilibrium flight path Black: Initial flight path angle = 0 Blue: plus increased initial airspeed Green: loop
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Flight of a Paper Airplane Example 1.3-1, Flight Dynamics
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Red: Equilibrium flight path Black: Initial flight path angle = 0 Blue: plus increased initial airspeed Green: loop
Assignment #2
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Compute the trajectory of a balsa glider
Configuration Aerodynamics Gliding Flight
Notation for Scalars and Vectors • Scalar: usually lower case: a , b , c , …, x , y , z a = 12;
b = 7;
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Matrices and Transpose Dimension = (m x n )
x
=
! p $ # & # q &; # r & " %
A
=
! # # # # #"
b e
g
h
k &
l
m
n &%
( 3 ! 1)
&
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" 2 % $ '; $ !7 ' $# 16 '&
ax
=
xa
=
! # # # "
ax
1
T =
! "
x1
x2
x3
# $
A
T
=
x
ax
2
ax
3
$ & & & %
• Transpose: interchange rows and columns
x
+
b
2
=
12 + 49 = 61
=
" $ $ $ #
x1 x2 x 3
% ' '; ' &
y
" $ $ $ $ #
=
a b c d
% ' ' ' ' &
Operands must be conformable Multiplication of vector by scalar is associative, commutative, and distributive
( 4 ! 3)
! a d g l $ # & # b e h m & # c f k n & " %
x = a
Multiplication
c $ & f &
a d
b = 19;
Ordered set Column of scalars Dimension = n x 1
a
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+
• Vector: usually bold or with underbar: x or x
Math Preliminaries
• Matrix: usually bold capital or capital: F or F
c = a
ax
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T =
a
( x + y ) = ( x + y ) a = ( ax + a y ) ( )
dim x
! "
Could we add ( x + a ) ?
ax1
•
=
( )
dim y
ax2
Only if
ax3
dim
# $
( x ) (1 ! 1) =
Inner Product
Addition • Conformable vectors and matrices are added term by term
x
=
! # "
a b
$ & %
z
;
=
! # "
c d
Inner (dot) product of vectors produces a scalar result
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$ & %
x
T
x
=
x
•
x
=
! "
! # "
a + c b + d
x2
=
2
2
$ & %
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Matrix-vector product transforms one vector into another Matrix-matrix product produces a new matrix
y Ax =
=
(n ! 1) ( n ! m )( m ! 1) =
x
1
x
2
x
3
d x % " ' $ ' $ '=$ ' $ ' $ '& #
x 3
)
2 1
2 2 1/2 2 + x 3
)
• Derivatives and integrals of vectors are vectors of derivatives and integrals
% ' ' ' & " ( 2 x + 4 x + 6 x ) 1 2 3 $ $ ( 3 x1 ! 5 x2 + 7 x 3 ) =$ $ ( 4 x1 + x 2 + 8 x3 ) $ $# ( !9 x1 ! 6 x2 ! 3 x3 )
x 2
# & & & $
Derivatives and Integrals of Vectors
Vector Transformation
%" '$ '$ '$ '# &
x1
Length (or magnitude) of vector is square root of dot product
(
" 2 4 6 $ $ 3 !5 7 $ 4 1 8 $ !9 !6 !3 #
2
= x 1 + x 2 + x 3
= x + x
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x3
(1 ! m)( m ! 1) (1 ! 1)
(
x + z =
x1
! % #% $ % "
y1 y2 y3 y4
% ' ' ' ' ' &
dt
=
! # # # # # #"
dx1 dt dx2 dt dx3 dt
$ & & & & & &%
!
x dt
=
" $ $ $ $ $ #
! x dt ! x dt ! x dt 1
2
3
% ' ' ' ' ' &
Matrix Inverse
Matrix Identity and Inverse •
x2
Transformation
! # # #"
x y z
$ & & &% 2
=
! # # #"
cos'
0
0
1
=
( sin' $ ! &# 0 &# cos' & % #"
sin' 0
x1
Inverse Transformation
=
A
Ax 1
$ & & &% 1
x y z
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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
!1
x2 AA
! # # #"
x y z
$ & & &% 1
=
! cos' # 0 # #" ( sin'
0
sin'
1
0
0
cos'
$! &# &# &% #"
x y z
!1
$ & & &% 2
=
# % % %$ =
Actuators Sensors
=
! # # #"
1
0
0
0
1
0
0
0
1
AA
!1
=
$ & & &%
!1
y Iy
A A
=
=
I
!1
! sin" & # cos" 0 ! sin" & (% ( 0 1 0 (% 0 ( sin" 0 c os" ( ' %$ sin" 0 cos" (' # cos" 0 ! sin" & # cos" 0 s in" & % (% ( 1 0 1 0 % 0 (% 0 ( %$ sin" 0 c os" (' %$ ! sin" 0 c os" (' cos"
0
0
1
=
Dynamic Systems
I3
# % % %$
1
0
0
0
1
0
0
0
1
& ( ( ('
Mathematical Models of Dynamic Systems are Differential Equations Continuous-time dynamic process: Vector Ordinary Differential Equation
(x) ( ) dim ( u ) dim ( w ) dim ( p ) dim
x! (t ) "
d x(t ) dt
=
dim f =
f [ x(t ), u(t ), w(t ), p(t ), t ]
=
=
( n ! 1) ( n ! 1) ( m ! 1) ( s ! 1) (l ! 1)
=
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Output Transformation 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
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Observation Process: Measurement may contain error or be incomplete y = output (error-free) z = measurement n = measurement error
All of these quantities are vectors
y ( t ) h[x (t ), u(t )] =
dim
( y ) ( r ! 1) ( ) ( r ! 1)
dim h
=
=
Measurement with Error dim
z(t ) = y (t ) + n (t )
dim
( z ) ( r ! 1) ( n ) ( r ! 1) =
=
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 Equations •
Nonlinear, Time-Varying
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• d x (t ) dt
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f [ x(t ), u(t ), w(t ), p(t ), t ]
d x(t ) dt
=
[
f x (t ), u(t ), w(t )
]
d x(t ) dt
= F (t )x(t ) +
d x(t ) dt
= F
G(t )u(t ) + L(t )w (t )
x (t ) + Gu (t ) + Lw (t )
Linear, Time-Varying – Small amplitude motions – Perturbations from a dynamic flight path
Nonlinear, Time-Invariant – Large amplitude motions – Negligible change in mass
– Large amplitude motions – Significant change in mass
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Linear, Time-Invariant – Small amplitude motions – Perturbations from an equilibrium flight path
Simplified Longitudinal Modes of Motion
Simplified Longitudinal Modes of Motion
Phugoid (Long-Period) Mode
Short-Period Mode
Airspeed
Flight Path Angle
Pitch Rate
Angle of Attack
Simplified Lateral Modes of Motion
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Airspeed
Flight Path Angle
Pitch Rate
Angle of Attack
Simplified Lateral Modes of Motion
Dutch-Roll Mode
Yaw Rate
Note change in time scale
Roll and Spiral Modes
Sideslip Angle Roll Rate
Roll Angle
Flight Dynamics Book and Computer Code •
All programs are accessible from the Flight Dynamics web page – http://www.princeton.edu/~stengel/FlightDynamics.html
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... 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
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Linear system analysis (MATLAB) – http://www.princeton.edu/~stengel/FDcodeC.html
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Paper airplane simulation (MATLAB) – http://www.princeton.edu/~stengel/PaperPlane.html
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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 online) – http://lib-terminal.princeton.edu/ejournals/by_title_zd.asp
• NACA/NASA and AIAA pubs – http://ntrs.nasa.gov/search.jsp
Performance analysis of a business jet aircraft (Excel) – http://www.princeton.edu/~stengel/Example261.xls
Primary Learning Objectives !
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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 aero dynamics , and methods for analysis of linear and nonlinear systems.
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Appreciation of the historical context within which past aircraft have been designed and operated, providing a soun d footing for the development of future aircraft.
More Learning Objectives !
Detailed evaluation of the linear and nonlinear flight characteristics of a specific aircraft type.
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Improved skills for presenting ideas, orally and on paper.
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Improved ability to analyze complex, integrated problems.
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Demonstratedcomputing 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.
