Aircraft Flight Dynamics
Robert Stengel, Princeton University, 2012"
Copyright 2012 by Robert Stengel. All rights reserved. For educational use only.!
/>! 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
/>– 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
/>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!
• !
• 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
( )
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 ) + G u(t ) + L w(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"
– />• 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)"
– />Helpful Resources"
• Web pages"
– />– />– />• Princeton University Engineering Library (paper and on-
line)"
– />• NACA/NASA and AIAA pubs"
– />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."