MAE331 Lecture

Aircraft Control Devices and Systems

Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2012"

Copyright 2012 by Robert Stengel. All rights reserved. For educational use only.!http://www.princeton.edu/~stengel/MAE331.html!

http://www.princeton.edu/~stengel/FlightDynamics.html!

•  Control surfaces"•  Control mechanisms"•  Flight control systems"

Design for Control"

•  Elevator/stabilator: pitch control"•  Rudder: yaw control"•  Ailerons: roll control"•  Trailing-edge flaps: low-angle lift control"

•  Leading-edge flaps/slats: High-angle lift control"

•  Spoilers: Roll, lift, and drag control"•  Thrust: speed/altitude control"

Critical Issues for Control"•  Effect of control surface deflections on aircraft motions"

–  Generation of control forces and rigid-body moments on the aircraft"–  Rigid-body dynamics of the aircraft"�  δE is an input for longitudinal motion"

θ =

Mechanical, Power-Boosted System"Grumman A-6!

McDonnell Douglas F-15!

Critical Issues for Control"•  Command and control of the control surfaces"

–  Displacements, forces, and hinge moments of the control mechanisms"

–  Dynamics of control linkages included in model"�  δE is a state for mechanical dynamics"

δ E =Control Surface Dynamics

and Aerodynamics�

Aerodynamic and Mechanical Moments on Control Surfaces"

•  Increasing size and speed of aircraft leads to increased hinge moments"

•  This leads to need for mechanical or aerodynamic reduction of hinge moments"

•  Need for aerodynamically balanced surfaces"

•  Elevator hinge moment"

Helevator =CHelevator

12ρV 2Sc

Aerodynamic and Mechanical Moments on Control Surfaces"

CHsurface

= CH δδ + CHδ

δ + CHαα + CHcommand

+ ...

CH δ: aerodynamic/mechanical damping moment

CHδ: aerodynamic/mechanical spring moment

CHα: floating tendency

CHcommand:pilot or autopilot input

•  Hinge-moment coefficient, CH"–  Linear model of dynamic effects"

Angle of Attack and Control Surface Deflection"

•  Horizontal tail at positive angle of attack"

•  Horizontal tail with elevator control surface"

•  Horizontal tail with positive elevator deflection"

Floating and Restoring Moments on a Control Surface"

•  Positive elevator deflection produces a negative (�restoring�) moment, Hδ, on elevator due to aerodynamic or mechanical spring"

•  Positive angle of attack produces negative moment on the elevator"•  With �stick free�, i.e., no opposing torques, elevator �floats� up due

to negative Hδ"

Dynamic Model of a Control Surface Mechanism"

δ − H δδ − Hδδ = Hαα + Hcommand + ...

mechanism dynamics = external forcing

•  Approximate control dynamics by a 2nd-order LTI system"

•  Bring all torques and inertias to right side"

δE =Helevator

Ielevator=CHelevator

12ρV 2Sc

Ielevator

= CH δEδE + CHδE

δE + CHαα + CHcommand

+ ...$%

&'

12ρV 2Sc

Ielevator≡ H δE

δE + HδEδE + Hαα + Hcommand + ...

Dynamic Model of a Control Surface Mechanism"

Ielevator = effective inertia of surface, linkages, etc.

H δE =∂ Helevator Ielevator( )

∂ δ; HδE =

∂ Helevator Ielevator( )∂δ

Hα =∂ Helevator Ielevator( )

∂α

•  Stability and control derivatives of the control mechanism"

Coupling of System Model and Control Mechanism Dynamics "

•  2nd-order model of control-deflection dynamics"–  Command input from cockpit"–  Forcing by aerodynamic effects"

•  Control surface deflection"•  Aircraft angle of attack and angular rates"

•  Short period approximation"•  Coupling with mechanism dynamics"

ΔxSP = FSPΔxSP +GSPΔuSP = FSPΔxSP +FδESPΔxδE

Δ qΔ α

$

%&&

'

())≈

Mq Mα

1 −Lα VN

$

%

&&&

'

(

)))

ΔqΔα

$

%&&

'

())+

MδE 0

−LδE VN0

$

%

&&&

'

(

)))

ΔδEΔ δE

$

%&

'

()

ΔxδE = FδEΔxδE +GδEΔuδE +FSPδEΔxSP

Δ δEΔ δE

#

$%%

&

'((≈

0 1HδE H δE

#

$%%

&

'((

ΔδEΔ δE

#

$%

&

'(+

0−HδE

#

$%%

&

'((ΔδEcommand +

0 0Hq Hα

#

$%%

&

'((

ΔqΔα

#

$%%

&

'((

Short Period Model Augmented by Control Mechanism Dynamics "

•  Augmented dynamic equation"

•  Augmented stability and control matrices"

FSP/δE =FSP FδE

SP

FSPδE FδE

"

#

$$

%

&

''=

Mq Mα MδE 0

1 −Lα VN−LδE VN

0

0 0 0 1Hq Hα HδE H δE

"

#

$$$$$$

%

&

''''''

ΔxSP ' =

ΔqΔαΔδEΔ δE

$

%

&&&&&

'

(

)))))

ΔxSP /δE = FSP /δEΔxSP /δE +GSP /δEΔδEcommand

State Vector!

GSP /δE =

000HδE

"

#

$$$$

%

&

''''

Roots of the Augmented Short Period Model "

•  Characteristic equation for short-period/elevator dynamics"

ΔSP/δE s( ) = sIn −FSP/δE =

s−Mq( ) −Mα −MδE 0

−1 s+ Lα VN( ) LδEVN

0

0 0 s −1−Hq −Hα −HδE s−H δE( )

= 0

ΔSP /δE s( ) = s2 + 2ζSPωnSPs +ωnSP

2( ) s2 + 2ζδEωnδEs +ωnδE

2( )Short Period" Control Mechanism"

Roots of the Augmented Short Period Model "

•  Coupling of the modes depends on design parameters"

MδE ,LδE VN,Hq , and Hα

•  Desirable for mechanical natural frequency > short-period natural frequency"

•  Coupling dynamics can be evaluated by root locus analysis"

Horn Balance"

CH ≈ CHαα + CHδE

δE + CHpilot input

•  Stick-free case"–  Control surface free to �float� "

CH ≈ CHαα + CHδE

δE

•  Normally "

CHα< 0 : reduces short-period stability

CHδE< 0 : required for mechanical stability

NACA TR-927, 1948!

Horn Balance"•  Inertial and aerodynamic

effects"•  Control surface in front of

hinge line"–  Increasing elevator

improves pitch stability, to a point "

•  Too much horn area"–  Degrades restoring moment "–  Increases possibility of

mechanical instability"–  Increases possibility of

destabilizing coupling to short-period mode"

€

CHα

Overhang or Leading-Edge Balance"•  Area in front of the

hinge line"•  Effect is similar to

that of horn balance"•  Varying gap and

protrusion into airstream with deflection angle"

CH ≈ CHαα + CHδ

δ + CHpilot input

NACA TR-927, 1948!

All-Moving Control Surfaces"•  Particularly effective at supersonic speed (Boeing

Bomarc wing tips, North American X-15 horizontal and vertical tails, Grumman F-14 horizontal tail)"

•  SB.4�s �aero-isoclinic� wing"•  Sometimes used for trim only (e.g., Lockheed L-1011

horizontal tail)"•  Hinge moment variations with flight condition"

Shorts SB.4!

Boeing !Bomarc!

North American X-15!

Grumman F-14!

Lockheed L-1011!

Control Surface Types�

Elevator"•  Horizontal tail and elevator

in wing wake at selected angles of attack"

•  Effectiveness of low mounting is unaffected by wing wake at high angle of attack"

•  Effectiveness of high-mounted elevator is unaffected by wing wake at low to moderate angle of attack"

Ailerons"•  When one aileron goes up, the other goes down"

–  Average hinge moment affects stick force"

Compensating Ailerons"•  Frise aileron"

–  Asymmetric contour, with hinge line at or below lower aerodynamic surface"

–  Reduces hinge moment"•  Cross-coupling effects can be adverse or

favorable, e.g. yaw rate with roll"–  Up travel of one > down travel of other to

control yaw effect"

Abzug & Larrabee, 2002!

Spoilers"•  Spoiler reduces lift, increases drag"

–  Speed control"•  Differential spoilers"

–  Roll control "–  Avoid twist produced by outboard

ailerons on long, slender wings"–  free trailing edge for larger high-lift

flaps"•  Plug-slot spoiler on P-61 Black

Widow: low control force"•  Hinged flap has high hinge moment"

North American P-61!


Elevons"•  Combined pitch and roll control

using symmetric and asymmetric surface deflection"

•  Principally used on"–  Delta-wing configurations"–  Swing-wing aircraft"

Grumman F-14!

General Dynamics F-106!

Canards"•  Pitch control"

–  Ahead of wing downwash"–  High angle of attack

effectiveness"–  Desirable flying qualities

effect (TBD)"

Dassault Rafale!

SAAB Gripen!

Yaw Control of Tailless Configurations"•  Typically unstable in pitch and yaw"•  Dependent on flight control system

for stability"•  Split ailerons or differential drag

flaps produce yawing moment"

McDonnell Douglas X-36!

Northrop Grumman B-2!

Rudder"•  Rudder provides yaw control"

–  Turn coordination"–  Countering adverse yaw"–  Crosswind correction"–  Countering yaw due to engine loss"

•  Strong rolling effect, particularly at high α"•  Only control surface whose nominal

aerodynamic angle is zero"•  Possible nonlinear effect at low deflection

angle"•  Insensitivity at high supersonic speed"

–  Wedge shape, all-moving surface on North American X-15"

Martin B-57!

Bell X-2!

Rudder Has Mechanical As Well as Aerodynamic Effects "

!  American Airlines 587 takeoff behind Japan Air 47, Nov. 12, 2001"!  Excessive periodic commands to rudder caused vertical tail failure"

Japan B-747!American A-300!

http://www.usatoday.com/story/travel/flights/2012/11/19/airbus-rudder/1707421/!

NTSB Simulation of American Flight 587 "

!  Flight simulation derived from digital flight data recorder (DFDR) tape"

Control Mechanization �Effects�

Control Mechanization Effects"

•  Fabric-covered control surfaces (e.g., DC-3, Spitfire) subject to distortion under air loads, changing stability and control characteristics"

•  Control cable stretching"•  Elasticity of the airframe

changes cable/pushrod geometry"

•  Nonlinear control effects"–  friction"–  breakout forces"–  backlash"

Douglas DC-3!

Supermarine !Spitfire!

Nonlinear Control Mechanism Effects"•  Friction"•  Deadzone"

Control Mechanization Effects"•  Breakout force"•  Force threshold"

B-52 Control Compromises to Minimize Required Control Power"

•  Limited-authority rudder, allowed by "–  Low maneuvering requirement "–  Reduced engine-out requirement (1 of

8 engines) "–  Crosswind landing gear"

•  Limited-authority elevator, allowed by "–  Low maneuvering requirement "–  Movable stabilator for trim"–  Fuel pumping to shift center of mass"

•  Small manually controlled "feeler" ailerons with spring tabs "

–  Primary roll control from powered spoilers, minimizing wing twist"

Internally Balanced Control Surface"

!  B-52 application"!  Control-surface fin

with flexible seal moves within an internal cavity in the main surface"

!  Differential pressures reduce control hinge moment"


δ + CHpilot input

Boeing B-52!

B-52 Rudder Control Linkages"B-52 Mechanical

Yaw Damper"•  Combined stable rudder tab, low-friction bearings, small

bobweight, and eddy-current damper for B-52"•  Advantages"

–  Requires no power, sensors, actuators, or computers"–  May involve simple mechanical components"

•  Problems"–  Misalignment, need for high precision"–  Friction and wear over time"–  Jamming, galling, and fouling"–  High sensitivity to operating conditions, design difficulty"

Boeing B-47 Yaw Damper"•  Yaw rate gyro drives rudder to increase

Dutch roll damping"•  Comment: �The plane wouldn�t need this

contraption if it had been designed right in the first place.�"

•  However, mode characteristics -- especially damping -- vary greatly with altitude, and most jet aircraft have yaw dampers"

•  Yaw rate washout to reduce opposition to steady turns"

Northrop YB-49 Yaw Damper!•  Minimal directional stability due to small vertical surfaces

and short moment arm"•  Clamshell rudders, like drag flaps on the B-2 Spirit"•  The first stealth aircraft, though that was not intended"•  Edwards AFB named after test pilot, Glen Edwards,

Princeton MSE, killed testing the aircraft"•  B-49s were chopped up after decision not to go into

production"•  Northrop had the last word: it built the B-2!

Northrop YB-49!

Northrop/Grumman B-2!

Northrop N-9M!

Instabilities Due To Control Mechanization"

•  Aileron buzz (aero-mechanical instability; P-80)"•  Rudder snaking (Dutch roll/mechanical coupling; Meteor, He-162)"•  Aeroelastic coupling (B-47, Boeing 707 yaw dampers)"

Rudder Snaking"•  Control-free dynamics"

–  Nominally symmetric control position"–  Internal friction"–  Aerodynamic imbalance"

•  Coupling of mechanical motion with Dutch roll mode"

Douglas DC-2!

•  Solutions"–  Trailing-edge bevel"–  Flat-sided surfaces"–  Fully powered controls"

Roll/Spiral Limit Cycle Due to Aileron Imbalance"

•  Unstable nonlinear oscillation grows until it reaches a steady state"

•  This is called a limit cycle"

Lockheed P-38!Control Surface Buzz"

North American FJ-4!

•  At transonic speed, normal shocks may occur on control surface"–  With deflection, shocks move

differentially "–  Possibility of self-sustained

nonlinear oscillation (limit cycle)"

ARC R&M 3364!

•  Solutions "–  Splitter-plate rudder

fixes shock location for small deflections"

–  Blunt trailing edge"–  Fully powered

controls with actuators at the surfaces"

Rudder Lock"•  Rudder deflected to stops at high

sideslip; aircraft trims at high α"•  3 necessary ingredients"

–  Low directional stability at high sideslip due to stalling of fin"

–  High (positive) hinge moment-due-to-sideslip at high sideslip (e.g., B-26)!

–  Negative rudder yawing moment "•  Problematical if rudder is

unpowered and requires high foot-pedal force (�rudder float� of large WWII aircraft)"

•  Solutions"–  Increase high-sideslip directional

stability by adding a dorsal fin (e.g., B-737-100 (before), B-737-400 (after))"

–  Hydraulically powered rudder"

Martin B-26!

Boeing 737-100!

Boeing 737-400!

Control Systems�

SAS = Stability Augmentation System!

Downsprings and Bobweights"•  Adjustment of "

–  Stick-free pitch trim moment"–  Stick-force sensitivity to

airspeed*"•  Downspring"

–  Mechanical spring with low spring constant"

–  Exerts a ~constant trailing-edge down moment on the elevator!

•  Bobweight"–  Similar effect to that of the

downspring"–  Weight on control column that

affects feel or basic stability"–  Mechanical stability augmentation

(weight is sensitive to aircraft’s angular rotation)"

Beechcraft B-18!

* See pp. 541-545, Section 5.5, Flight Dynamics!

Effect of Scalar Feedback Control on Roots of the System "

Δy(s)= H (s)Δu(s)= kn(s)d(s)

Δu(s)= kn(s)d(s)

KΔε(s)

•  �Block diagram algebra�"

H (s)= kn(s)d(s)

= KH (s) Δyc(s)−Δy(s)[ ]Δy(s)= KH (s)Δyc(s)−KH (s)Δy(s)

K

Closed-Loop Transfer Function "

1+KH (s)[ ]Δy(s)= KH (s)Δyc(s)

Δy(s)Δyc(s)

=KH (s)1+KH (s)[ ]

Roots of the Closed-Loop System "

•  Closed-loop roots are solutions to"

Δclosedloop

(s) = d(s) + Kkn(s) = 0

or! K kn(s)d(s)

= −1

Δy(s)Δyc(s)

=K kn(s)d(s)

1+K kn(s)d(s)

"

#$

%

&'

=Kkn(s)

d(s)+Kkn(s)[ ]=Kkn(s)Δclosedloop

s( )

Root Locus Analysis of Pitch Rate Feedback to Elevator (2nd-Order Approximation)"

KH s( ) = K Δq(s)ΔδE(s)

= Kkq s− zq( )

s2 +2ζSPωnSPs+ωnSP

2 = −1

!  # of roots = 2"!  # of zeros = 1!!  Destinations of roots (for k =

±∞):"!  1 root goes to zero of n(s)"!  1 root goes to infinite radius"

!  Angles of asymptotes, θ, for the roots going to ∞"!  K -> +∞: –180 deg"!  K -> –∞: 0 deg"

Root Locus Analysis of Pitch Rate Feedback to Elevator (2nd-Order Approximation)"

•  �Center of gravity� : doesn�t matter"

•  Locus on real axis"–  K > 0: Segment to the left of

the zero"–  K < 0: Segment to the right of

the zero"

Feedback effect is analogous to changing Mq "

Root Locus Analysis of Angular Feedback to Elevator (4th-Order Model)*"

Flight Path Angle! Pitch Rate!

Pitch Angle! Angle of Attack!

* p. 524, Flight Dynamics"

Root Locus Analysis of Angular Feedback to Thrust (4th-Order Model)"

Flight Path Angle! Pitch Rate!

Pitch Angle! Angle of Attack!

Direct Lift and Propulsion Control�

Direct-Lift Control-Approach Power Compensation"

•  F-8 Crusader "–  Variable-incidence wing,

better pilot visibility"–  Flight path control at low

approach speeds "•  requires throttle use "•  could not be accomplished

with pitch control alone "–  Engine response time is slow"–  Flight test of direct lift control

(DLC), using ailerons as flaps"•  Approach power

compensation for A-7 Corsair II and direct lift control studied using Princeton’s Variable-Response Research Aircraft"

Princeton VRA!

Vought A-7!

Vought F-8!

Direct-Lift/Drag Control"

•  Direct-lift control on S-3A Viking"–  Implemented with spoilers"–  Rigged �up� during landing

to allow ± lift."•  Speed brakes on T-45A

Goshawk make up for slow spool-up time of jet engine"–  BAE Hawk's speed brake

moved to sides for carrier landing"

–  Idle speed increased from 55% to 78% to allow more effective modulation via speed brakes"

Lockheed S-3A!

Boeing T-45!

Next Time:�Flight Testing for �

Stability and Control��

Reading�Flight Dynamics, 419-428 �

Aircraft Stability and Control, Ch. 3 �Virtual Textbook, Part 17 �

Supplementary!Material!

Trailing-Edge Bevel Balance"

•  Bevel has strong effect on aerodynamic hinge moments"

•  See discussion in Abzug and Larrabee!


δ + CHpilot input

Control Tabs"•  Balancing or geared tabs"

–  Tab is linked to the main surface in opposition to control motion, reducing the hinge moment with little change in control effect"

•  Flying tabs"–  Pilot's controls affect only the

tab, whose hinge moment moves the control surface"

•  Linked tabs"–  divide pilot's input between tab

and main surface"•  Spring tabs "

–  put a spring in the link to the main surface"

Control Flap Carryover Effect on Lift Produced By Total Surface"

from Schlichting & Truckenbrodt!

CLδE

CLα

vs.cf

x f + cf

€

c f x f + c f( )

Aft Flap vs. All-Moving Control Surface"

•  Carryover effect"–  Aft-flap deflection can be almost as effective as

full surface deflection at subsonic speeds"–  Negligible at supersonic speed"

•  Aft flap "–  Mass and inertia lower, reducing likelihood of

mechanical instability"–  Aerodynamic hinge moment is lower"–  Can be mounted on structurally rigid main

surface"

Mechanical and Augmented Control Systems"

•  Mechanical system"–  Push rods, bellcranks, cables, pulleys"

•  Power boost"–  Pilot's input augmented by hydraulic servo that

lowers manual force"•  Fully powered (irreversible) system"

–  No direct mechanical path from pilot to controls"

–  Mechanical linkages from cockpit controls to servo actuators"

"

Boeing 767 Elevator Control System"


Boeing 777 Fly-By-Wire Control System"

�Classical� Lateral Control Logic for a Fighter Aircraft (c.1970)"

MIL-DTL-9490E, Flight Control Systems - Design, Installation and Test of Piloted Aircraft, General Specification for, 22 April 2008"

Superseded for new designs on same date by"SAE-AS94900"

http://www.sae.org/servlets/works/documentHome.do?comtID=TEAA6A3&docID=AS94900&inputPage=dOcDeTaIlS!

The Unpowered F4D Rudder"•  Rudder not a problem under normal flight conditions"

–  Single-engine, delta-wing aircraft requiring small rudder inputs"•  Not a factor for upright spin "

–  Rudder was ineffectual, shielded from flow by the large delta wing"•  However, in an inverted spin "

–  rudder effectiveness was high "–  floating tendency deflected rudder in a pro-spin direction "–  300 lb of pedal force to neutralize the rudder"

•  Fortunately, the test aircraft had a spin chute"

Powered Flight Control Systems"•  Early powered systems had a single

powered channel, with mechanical backup"

–  Pilot-initiated reversion to "conventional" manual controls"

–  Flying qualities with manual control often unacceptable"

•  Reversion typically could not be undone"

–  Gearing change between control stick and control to produce acceptable pilot load"

–  Flying qualities changed during a high-stress event"

•  Hydraulic system failure was common"–  Redundancy was needed"

•  Alternative to eject in military aircraft"

A4D!

A3D!

B-47!

Advanced Control Systems"•  Artificial-feel system"

– Restores control forces to those of an "honest" airplane"

–  "q-feel" modifies force gradient"– Variation with trim stabilizer angle"– Bobweight responds to gravity and to

normal acceleration"•  Fly-by-wire/light system"

– Minimal mechanical runs"– Command input and feedback signals

drive servo actuators"– Fully powered systems"– Move from hydraulic to electric power"

Control-Configured Vehicles"•  Command/stability augmentation"•  Lateral-directional response"

–  Bank without turn"–  Turn without bank"–  Yaw without lateral translation"–  Lateral translation without yaw"–  Velocity-axis roll (i.e., bank)"

•  Longitudinal response"–  Pitch without heave"–  Heave without pitch"–  Normal load factor"–  Pitch-command/attitude-hold"–  Flight path angle"

USAF F-15 IFCS!

Princeton Variable-Response Research Aircraft!USAF AFTI/F-16!

United Flight 232, DC-10Sioux City, IA, 1989"

•  Uncontained engine failure damaged all three flight control hydraulic systems (http://en.wikipedia.org/wiki/United_Airlines_Flight_232)"

United Flight 232, DC-10Sioux City, IA, 1989"

•  Pilot maneuvered on differential control of engines to make a runway approach"•  101 people died"•  185 survived"

Propulsion Controlled Aircraft"•  Proposed backup attitude control in event of flight control system failure"•  Differential throttling of engines to produce control moments"•  Requires feedback control for satisfactory flying qualities"

NASA MD-11 PCA Flight Test!

NASA F-15 PCA Flight Test!

Proposed retrofit to McDonnell-Douglas (Boeing) C-17!

Stability-and-Control Flight Testing

Robert Stengel, Aircraft Flight DynamicsMAE 331, 2012"



•  Flight test instrumentation"•  Pilot opinion ratings"•  Flying qualities requirements"•  Flying qualities specifications"•  Pilot-induced oscillations"

Flying (or Handling) Qualities"•  Stability and controllability

perceived by the pilot"•  1919 flight tests of Curtiss

JN-4H Jenny at NACA Langley Laboratory by Warner, Norton, and Allen"–  Elevator angle and stick

force for equilibrium flight"–  Correlation of elevator angle

and airspeed with stability"–  Correlation of elevator angle

and airspeed with wind tunnel tests of pitch moment"

Early Flight Testing Instrumentation"•  Flight recording instruments: drum/strip charts, inked needles, film,

galvanometers connected to air vanes, pressure sensors, clocks"

Hundreds/Thousands of Measurements Made in Modern Flight Testing"

Modern Approach to Flight Testing Instrumentation"

iPhone"•  3-axis accelerometer"•  3-axis angular rate"•  2-axis magnetometer

compass"•  GPS position

measurement"•  1 GHz processor"•  512 MB RAM"•  32 GB flash memory"

z =

uvwpqr

εhorizontalεverticalLλh

#

$

%%%%%%%%%%%%%%%

&

'

(((((((((((((((

First Flying Qualities Specification"!  First flying qualities specification: 1935 "

!  Edward Warner. Douglas DC-4 transport "!  Interviews with pilots and engineers"

Flying Qualities Research at NACA"•  Hartley Soulé and Floyd Thompson

(late 1930s)"–  Long- and short-period motions"–  Time to reach specified bank angle"–  Period and damping of oscillations"–  Correlation with pilot opinion"

•  Robert Gilruth (1941-3)"–  Parametric regions and boundaries"–  Multi-aircraft criteria"–  Control deflection, stick force, and

normal load factor"–  Roll helix angle"–  Lateral control power"

Gilruth Roll-Rate Criterion [pb/2V]"•  Helix angle formed by

rotating wing tips, pb/2V!–  Roll rate, p, rad/s"–  Wing semi-span, b/2, m"–  Velocity, V, m/s"

•  Robert Gilruth criterion"–  pb/2V > 0.07 rad"

NACA TR-715, 1941!

Simplified Roll-Rate Response "•  Tradeoff between high pb/2V and

high lateral stick forces prior to powered controls:"

p(t) = p(0)eat

p(t) = [Clpp(t) + ClδA

δA(t)]qSb / Ixx= a p(t) + cδA(t)

p(t)= caeat −1( )δAstep

pSS = −ClδA

Clp

δASS

•  Initial-condition response (δA = 0) "

•  Step response [p(0) = 0]"

•  Steady-state response"

NACA TR-868!

IAS, mph"

pSSmax ,° / sec

Aircraft That Simulate Other Aircraft"•  Closed-loop control"•  Variable-stability research aircraft, e.g., TIFS, AFTI

F-16, NT-33A, and Princeton Variable-Response Research Aircraft (Navion)"

USAF/Calspan TIFS!USAF AFTI F-16!

Princeton VRA!USAF/Calspan NT-33A!

Cooper-Harper Handling Qualities Rating Scale"

NASA TN-D-5153,1969!

Effect of Equivalent Time Delay on Cooper-Harper Rating"

Short-Period �Bullseye� or �Thumbprint�"

ζSP

ωnSP

Carrier Approach on Back Side of the Power/Thrust Curve"

•  Precise path and airspeed control while on the back side of the power curve"–  Slower speed requires higher thrust"–  Lightly damped phugoid mode requires

"coordination of pitch and thrust control"•  Reference flight path generated by optical

device, which projects a meatball relative to a datum line"

Pilot-Induced Oscillations"•  MIL-F-8785C specifies no tendency for pilot-induced

oscillations (PIO)"–  Uncommanded aircraft is stable but piloting actions couple

with aircraft dynamics to produce instability"F-22! Space Shuttle!

Pilot-Induced Oscillations"•  Category I: Linear pilot-vehicle system oscillations"•  Category II: Quasilinear events with nonlinear contributions"•  Category III: Nonlinear oscillations with transients!

Hodgkinson, Neal, Smith, Geddes, Gibson et al!

NASA DFBW F-8 Simulation of Space Shuttle!

YF-16 Test Flight Zero"•  High-speed taxi test; no flight intended!•  Pilot-induced oscillations from sensitive roll control"•  Tail strike"•  Pilot elected to go around rather than eject"

Military Flying Qualities Specifications, MIL-F-8785C"

•  Specifications established during WWII "•  US Air Force and Navy coordinated efforts

beginning in 1945"•  First version appeared in 1948, last in 1980"•  Distinctions by flight phase, mission, and aircraft

type"•  Replaced by Military Flying Qualities Standard,

MIL-STD-1797A, with procurement-specific criteria"

MIL-F-8785C Aircraft Types"I.  Small, light airplanes, e.g., utility aircraft and

primary trainers"II.  Medium-weight, low-to-medium

maneuverability airplanes, e.g., small transports or tactical bombers"

III.  Large, heavy, low-to-medium maneuverability airplanes, e.g., heavy transports, tankers, or bombers"

IV.  Highly maneuverable aircraft, e.g., fighter and attack airplanes"

MIL-F-8785C Flight Phase"A.   Non-terminal flight requiring rapid maneuvering precise

tracking, or precise flight path control "•  air-to-air combat "•  ground attack "•  in-flight refueling (receiver) "•  close reconnaissance "•  terrain following "•  close formation flying"

B.   Non-terminal flight requiring gradual maneuvering"•  climb, cruise "•  in-flight refueling (tanker) "•  descent"

C.   Terminal flight "•  takeoff (normal and catapult) "•  approach "•  wave-off/go-around "•  landing"

MIL-F-8785C Levels of Performance"1.   Flying qualities clearly adequate for the mission

flight phase"2.   Flying qualities adequate to accomplish the

mission flight phase, with some increase in pilot workload or degradation of mission effectiveness"

3.   Flying qualities such that the aircraft can be controlled safely, but pilot workload is excessive or mission effectiveness is inadequate"

Principal MIL-F-8785C Metrics"•  Longitudinal flying

qualities"–  static speed stability"–  phugoid stability"–  flight path stability"–  short period frequency

and its relationship to command acceleration sensitivity"

–  short period damping"–  control-force gradients"

•  Lateral-directional flying qualities"–  natural frequency and damping

of the Dutch roll mode"–  time constants of the roll and

spiral modes"–  rolling response to commands

and Dutch roll oscillation"–  sideslip excursions"–  maximum stick and pedal forces"–  turn coordination"

Next Time:�Advanced Problems of Longitudinal Dynamics�

�

Reading�Flight Dynamics, 204-206, 503-525 �

Aircraft Stability and Control, Ch. 13 �Virtual Textbook, Part 18 �

Supplementary Material

Princeton University�s Flight Research Laboratory (1943-1983)


•  Forrestal Campus"•  3,000-ft dedicated runway"



Helicopters and Flying Saucers"

•  Piasecki HUP-1 helicopter!•  Hiller H-23 helicopter!•  Princeton Air Scooter!•  Hiller VZ-1 Flying Platform!•  Princeton 20-ft Ground Effect Machine"

Short-Takeoff-and-Landing, Inflatable Plane, and the Princeton Sailwing"

•  Pilatus Porter !•  Goodyear InflatoPlane!•  Princeton Sailwing"

Variable-Response Research Aircraft(Modified North American Navion A)"

Avionics Research Aircraft(Modified Ryan Navion A)"

Navion in the NASA Langley Research Center 30� x 60� Wind Tunnel"

Lockheed LASA-60 Utility Aircraft" Schweizer 2-32 Sailplane (�Cibola�)"

Steve Sliwa, �77, landing on Forrestal Campus runway.!currently CEO, In Situ, Inc.!

Apple iPhone Used for On-Board Data Processing and Recording

Jillian Alfred, Clayton Flanders, Brendan MahonPrinceton Senior Project, 2010"

iPhone Installation! Hobbico NexSTAR!

System Components! Pitot Tube Placement!

Autonomous UAV Control in a Simulated Air Traffic Control SystemAtray Dixit, Jaiye Falusi, Samuel Kim, Gabriel SavitPrinceton Senior Project, 2012"

Overview! System Hardware!

Aerial Refueling"•  Difficult flying task"•  High potential for PIO"•  Alternative designs"

–  Rigid boom (USAF)"–  Probe and drogue (USN)"

Formation Flying"•  Coordination and precision"•  Potential aerodynamic interference"•  US Navy Blue Angels (F/A-18)"

MIL-F-8785C Superseded by MIL-STD-1797"

•  Handbook for guidance rather than a requirement"•  Body of report is a form, with numbers to be filled in for

each new aircraft, e.g.,"

•  Useful reference data contained in Appendix A (~700 pages)"

UAV Handling Qualities"•  UAV Handling Qualities.....You Must Be Joking,

Warren Williams, 2003"–  UAV missions are diverse and complex"–  All UAVs must have sophisticated closed-loop flight

control systems"–  Cockpit is on the ground; significant time delays"–  Launch and recovery different from takeoff and landing"

•  Suggestion: Follow the form of MIL-F-8785C, FAR Part 23, etc., but adapt to differences between manned and unmanned systems"

Flight Testing for Certification in Other Agencies"

•  Federal Aviation Administration Airworthiness Standards"–  Part 23: GA"–  Part 25: Transports"

•  UK Civil Aviation Authority"•  European Aviation Safety Agency"•  Transport Canada"

Even the Best Specs Cannot Prevent Pilot Error"

On September 24, 1994, a TAROM Airbus A310, Flight 381, from Bucharest on approach to Paris Orly went into a sudden and uncommanded nose-up position and stalled. The crew attempted to countermand the plane's flight control system but were unable to get the nose down while remaining on course. Witnesses saw the plane climb to a tail stand, then bank sharply left, then right, then fall into a steep dive. Only when the dive produced additional speed was the crew able to recover steady flight. !!An investigation found that an overshoot of flap placard speed during approach, incorrectly commanded by the captain, caused a mode transition to flight level change. The auto-throttles increased power and trim went full nose-up as a result. The crew attempt at commanding the nose-down elevator could not counteract effect of stabilizer nose-up trim, and the resulting dive brought the plane from a height of 4100 feet at the time of the stall to 800 feet when the crew was able to recover command. !!The plane landed safely after a second approach. There were 186 people aboard. [Wikipedia]!

TAROM Flight 381 (A310 �Muntenia�)!http://www.youtube.com/watch?v=VqmrRFeYzBI!

Pilot Error, or Aircraft Maintenance, or Both?"

TAROM Flight 371 (A310 �Muntenia�)!http://www.youtube.com/watch?v=htzv2KebEkI&NR=1&feature=fvwp!

TAROM Flight 371 was an Airbus A310 that crashed near Baloteşti in Romania on 31 March 1995. It was a flight from Bucharest's main Otopeni airport to Brussels. The flight crashed shortly after it took off. Two main reasons are indicated: first the throttle of the starboard engine jammed, remaining in takeoff thrust, while the other engine reduced slowly to idle, creating an asymmetrical thrust condition that ultimately caused the aircraft to roll over and crash. Second, the crew failed to respond to the thrust asymmetry.!!None of the 10 crew and 50 passengers survived. [Wikipedia]!

Point-Mass Dynamics and Aerodynamic/Thrust Forces

Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2012!

•  Properties of the Atmosphere"•  Frames of reference"•  Velocity and momentum"•  Newton�s laws"•  Introduction to Lift, Drag, and Thrust"•  Simplified longitudinal equations of

motion"



The Atmosphere �

•  Air density and pressure decay exponentially with altitude"

•  Air temperature and speed of sound are linear functions of altitude "

Properties of the Lower Atmosphere! Wind: Motion of the Atmosphere"

•  Zero wind at Earth�s surface = Inertially rotating air mass"•  Wind measured with respect to Earth�s rotating surface "

Wind Velocity Profiles vary over Time!

Typical Jetstream Velocity!

•  Airspeed = Airplane�s speed with respect to air mass"•  Inertial velocity = Wind velocity ± Airspeed "

Air Density, Dynamic Pressure, and Mach Number"

ρ = Air density, functionof height= ρsealevele

βz = ρsealevele−βh

ρsealevel =1.225 kg /m3; β =1/ 9,042m

Vair = vx2 + vy

2 + vz2!" #$air1/2= vTv!" #$air

1/2= Airspeed

Dynamic pressure = q = 12ρ h( )Vair2

Mach number = Vaira h( )

; a = speed of sound, m / s •  Airspeed must increase as altitude increases to maintain constant dynamic pressure"

Contours of Constant Dynamic Pressure, "

Weight = Lift =CL12ρVair

2 S =CLqS•  In steady, cruising flight, "

q

Equations of Motion for a Point Mass�

Newtonian Frame of Reference"•  Newtonian (Inertial) Frame of

Reference"–  Unaccelerated Cartesian frame

whose origin is referenced to an inertial (non-moving) frame"

–  Right-hand rule"–  Origin can translate at constant

linear velocity"–  Frame cannot be rotating with

respect to inertial origin"

•  Translation changes the position of an object"

r =xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

•  Position: 3 dimensions"•  What is a non-moving frame?"

Velocity and Momentum "•  Velocity of a particle"

v = dxdt

= x =xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

vxvyvz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

•  Linear momentum of a particle"

p = mv = mvxvyvz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

where m = mass of particle

Newton�s Laws of Motion: Dynamics of a Particle "

•  First Law"–  If no force acts on a particle, it remains at rest or

continues to move in a straight line at constant velocity, as observed in an inertial reference frame -- Momentum is conserved"

ddt

mv( ) = 0 ; mv t1= mv t2


ddt

mv( ) = m dvdt

= F ; F =fxfyfz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

•  Second Law"–  A particle of fixed mass acted upon by a force

changes velocity with an acceleration proportional to and in the direction of the force, as observed in an inertial reference frame; "

–  The ratio of force to acceleration is the mass of the particle: F = m a"

∴dvdt

=1mF =

1mI3F =

1 / m 0 00 1 / m 00 0 1 / m

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

fxfyfz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


•  Third Law"–  For every action, there is an equal and opposite reaction"

Equations of Motion for a Point Mass: Position and Velocity "

dvdt

= v =vxvyvz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=1mF =

1 / m 0 00 1 / m 00 0 1 / m

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

fxfyfz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

drdt

= r =xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= v =

vxvyvz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

FI =fxfyfz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥ I

= Fgravity + Faerodynamics + Fthrust⎡⎣ ⎤⎦ I

Rate of change of position!

Rate of change of velocity!

Vector of combined forces!

Equations of Motion for a Point Mass "

�

˙ x (t) = dx(t)dt

= f[x(t),F]

•  Written as a single equation"

x ≡ rv

"

#$

%

&'=

PositionVelocity

"

#$$

%

&''=

xyzvxvyvz

"

#

$$$$$$$$

%

&

''''''''

•  With"

xyzvxvyvz

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

vxvyvzfx / mfy / m

fz / m

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

=

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

xyzvxvyvz

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

+

0 0 00 0 00 0 0

1 / m 0 00 1 / m 00 0 1 / m

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

fxfyfz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Dynamic equations are linear!

�

˙ x (t) = dx(t)dt

= f[x(t),F]

Equations of Motion for a Point Mass !

Gravitational Force:Flat-Earth Approximation"

•  g is gravitational acceleration"•  mg is gravitational force!•  Independent of position!•  z measured down"

Fgravity( )I = Fgravity( )E = mg f = m00go

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

•  Approximation"–  Flat earth reference is an inertial

frame, e.g.,"•  North, East, Down"•  Range, Crossrange, Altitude (–)"

go 9.807 m / s2 at earth's surface

Aerodynamic Force"

FI =XYZ

!

"

###

$

%

&&&I

=

CX

CY

CZ

!

"

###

$

%

&&&I

12ρVair

2 S

=

CX

CY

CZ

!

"

###

$

%

&&&I

q S

•  Referenced to the Earth not the aircraft"

Inertial Frame" Body-Axis Frame" Velocity-Axis Frame"

FB =CX

CY

CZ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥B

q S FV =CD

CY

CL

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥q S

•  Aligned with the aircraft axes"

•  Aligned with and perpendicular to the direction of motion"

Non-Dimensional Aerodynamic Coefficients"

Body-Axis Frame" Velocity-Axis Frame"

CX

CY

CZ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥B

=axial force coefficientside force coefficient

normal force coefficient

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

CD

CY

CL

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

drag coefficientside force coefficient

lift coefficient

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

•  Functions of flight condition, control settings, and disturbances, e.g., CL = CL(δ, M, δE)"

•  Non-dimensional coefficients allow application of sub-scale model wind tunnel data to full-scale airplane"

u(t) :axial velocityw(t) :normal velocityV t( ) : velocity magnitudeα t( ) :angle of attackγ t( ) : flight path angleθ(t) : pitch angle

•  along vehicle centerline!•  perpendicular to centerline!•  along net direction of flight!•  angle between centerline and direction of flight!•  angle between direction of flight and local horizontal!•  angle between centerline and local horizontal!

Longitudinal Variables"

γ = θ −α (with wingtips level)

Lateral-Directional Variables"

β(t) : sideslip angleψ (t) : yaw angleξ t( ) :heading angleφ t( ) : roll angle

•  angle between centerline and direction of flight!•  angle between centerline and local horizontal!•  angle between direction of flight and compass reference

(e.g., north)!•  angle between true vertical and body z axis!

ξ =ψ + β (with wingtips level)

Introduction to Lift and Drag �

Lift and Drag are Oriented to the Velocity Vector"

•  Drag components sum to produce total drag"–  Skin friction"–  Base pressure differential"–  Shock-induced pressure differential (M > 1)"

•  Lift components sum to produce total lift"–  Pressure differential between upper and lower surfaces"

–  Wing"–  Fuselage"–  Horizontal tail"

Lift =CL12ρVair

2 S ≈ CL0+∂CL

∂αα

%

&'(

)*12ρVair

2 S

Drag =CD12ρVair

2 S ≈ CD0+εCL

2$% &'12ρVair

2 S

Aerodynamic Lift"

•  Fast flow over top + slow flow over bottom = Mean flow + Circulation"

•  Speed difference proportional to angle of attack"•  Kutta condition (stagnation points at leading and

trailing edges)"

Chord Section!

Streamlines!

Lift =CL12ρVair

2 S ≈ CLwing+CLfuselage

+CLtail( ) 12 ρVair2 S ≈ CL0

+∂CL

∂αα

%

&'(

)*qS

2D vs. 3D Lift"

•  Inward flow over upper surface"•  Outward flow over lower surface"•  Bound vorticity of wing produces tip vortices"

Inward-Outward Flow!

Tip Vortices!

2D vs. 3D Lift"Identical Chord Sections!

Infinite vs. Finite Span!

•  Finite aspect ratio reduces lift slope"

What is aspect ratio?!

Aerodynamic Drag"

•  Drag components"–  Parasite drag (friction, interference, base pressure

differential)"–  Induced drag (drag due to lift generation)"–  Wave drag (shock-induced pressure differential)"

•  In steady, subsonic flight"–  Parasite (form) drag

increases as V2"–  Induced drag proportional

to 1/V2"–  Total drag minimized at

one particular airspeed"

Drag =CD12ρVair

2 S ≈ CDp+CDi

+CDw( ) 12 ρVair2 S ≈ CD0

+εCL2$% &'qS

2-D Equations of Motion�

2-D Equations of Motion for a Point Mass"

xzvxvz

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

vxvzfx / mfz / m

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

0 0 1 00 0 0 10 0 0 00 0 0 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

xzvxvz

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

+

0 00 0

1 / m 00 1 / m

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

fxfz

⎡

⎣⎢⎢

⎤

⎦⎥⎥

•  Restrict motions to a vertical plane (i.e., motions in y direction = 0)"

•  Assume point mass location coincides with center of mass"

Transform Velocity from Cartesian to Polar Coordinates"

xz

⎡

⎣⎢

⎤

⎦⎥ =

vxvz

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

V cosγ−V sinγ

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⇒Vγ

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

x2 + z2

− sin−1 zV

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=vx2 + vz

2

− sin−1 vzV

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Vγ

⎡

⎣⎢⎢

⎤

⎦⎥⎥=ddt

vx2 + vz

2

− sin−1 vzV

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

ddt

vx2 + vz

2

−ddtsin−1 vz

V⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

•  Inertial axes -> wind axes and back"

•  Rates of change of velocity and flight path angle"

Longitudinal Point-Mass Equations of Motion"

r(t)= x(t)= vx =V (t)cosγ (t)h(t)= − z(t)= −vz =V (t)sinγ (t)

V (t)=Thrust −Drag−mg h( ) sinγ (t)

m=CT −CD( ) 1

2ρ h( )V 2 (t)S −mg h( ) sinγ (t)

m

γ (t)=Lift −mg h( )cosγ (t)

mV (t)=CL12ρ h( )V 2 (t)S −mg h( )cosγ (t)

mV (t)

r = rangeh = height (altitude)V = velocityγ = flight path angle

•  Equations of motion, assuming mass is fixed, thrust is aligned with the velocity vector, and windspeed = 0"

•  In steady, level flight"•  Thrust = Drag"•  Lift = Weight"

Introduction to Propulsion�

Reciprocating (Internal Combustion) Engine (1860s)"

•  Linear motion of pistons converted to rotary motion to drive propeller"

Single Cylinder"Turbo-Charger (1920s)"

•  Increases pressure of incoming air"

•  Thrust produced directly by exhaust gas"

Axial-flow Turbojet (von Ohain, Germany)!

Centrifugal-flow Turbojet (Whittle, UK)!

Turbojet Engines (1930s)"

Turboprop Engines (1940s)"

•  Exhaust gas drives a propeller to produce thrust"

•  Typically uses a centrifugal-flow compressor"

Turbojet + Afterburner (1950s)"

•  Fuel added to exhaust"•  Additional air may be introduced"

•  Dual rotation rates, N1 and N2, typical"

Turbofan Engine (1960s)"

•  Dual or triple rotation rates"

High Bypass Ratio Turbofan"

Propfan Engine!

Aft-fan Engine!

Ramjet and Scramjet"Ramjet (1940s)" Scramjet (1950s)"

Talos!

X-43!Hyper-X!

Thrust and Thrust Coefficient"

Thrust ≡ CT12ρV 2S

•  Non-dimensional thrust coefficient, CT!–  CT is a function of power/throttle

setting, fuel flow rate, blade angle, Mach number, ..."

•  Reference area, S, may be aircraft wing area, propeller disk area, or jet exhaust area"

Isp =Thrustm go

Specific Impulse, Units = m/sm/s2 = sec

m ≡Mass flow rate of on−board propellantgo ≡Gravitational acceleration at earth 's surface

Thrust and Specific Impulse"

Sensitivity of Thrust to Airspeed"Nominal Thrust = TN ≡ CTN

12ρVN

2S

•  If thrust is independent of velocity (= constant)"

∂T∂V

= 0 = ∂CT

∂V12ρVN

2S+CTNρVNS

.( )N = Nominal or reference( ) value

•  Turbojet thrust is independent of airspeed over a wide range"

∂CT

∂V= −CTN

/VN

Power"•  Assuming thrust is aligned with airspeed vector"

Power = P = Thrust ×Velocity ≡ CT12ρV 3S

•  If power is independent of velocity (= constant)"∂P∂V

= 0 = ∂CT

∂V12ρVN

3S+ 32CTN

ρVN2S

•  Velocity-independent power is typical of propeller-driven propulsion (reciprocating or turbine engine, with constant RPM or variable-pitch prop)"

∂CT

∂V= −3CTN

/VN

Next Time: �Aviation History �

�Reading �

Airplane Stability and Control, Ch. 1!Virtual Textbook, Part 3 �

Supplementary Material

Early Reciprocating Engines"•  Rotary Engine:"

–  Air-cooled"–  Crankshaft fixed"–  Cylinders turn with propeller"–  On/off control: No throttle"

Sopwith Triplane!

SPAD S.VII!•  V-8 Engine:"

–  Water-cooled"–  Crankshaft turns with propeller"

Reciprocating Engines"•  Rotary"

•  In-Line"

•  V-12" •  Opposed"

•  Radial"

Turbo-compound Reciprocating Engine"•  Exhaust gas drives the turbo-compressor"•  Napier Nomad II shown (1949)"

Jet Engine Nacelles"

Pulsejet"Flapper-valved motor (1940s)" Dynajet Red Head (1950s)"

V-1 Motor!Pulse Detonation Engine"

on Long EZ (1981)"

http://airplanesandrockets.com/motors/dynajet-engine.htm!

Fighter Aircraft and Engines"Lockheed P-38!

Allison V-1710!Turbocharged Reciprocating Engine!

Convair/GD F-102!

P&W J57!Axial-Flow Turbojet Engine!

MD F/A-18!

GE F404!Afterburning Turbofan Engine!

SR-71: P&W J58 Variable-Cycle Engine (Late 1950s)"

Hybrid Turbojet/Ramjet"

Flying Qualities CriteriaRobert Stengel, Aircraft Flight Dynamics

MAE 331, 2012"



•  MIL-F-8785C criteria!•  CAP, C*, and other longitudinal

criteria"•  ϕ/β, ωϕ/ω�, and other lateral-

directional criteria"•  Pilot-vehicle interactions"•  Flight control system design"

Design for Satisfactory Flying Qualities"•  Satisfy procurement requirement (e.g., Mil

Standard)"•  Satisfy test pilots (e.g., Cooper-Harper ratings)"•  Avoid pilot-induced oscillations (PIO)"•  Minimize time-delay effects"•  Time- and frequency-domain criteria"

MIL-F-8785C Identifies Satisfactory, Acceptable, and Unacceptable Response Characteristics"

Damping Ratio"

Step Response"

Frequency Response"

Short-period angle-of-attack response to elevator input!

Longitudinal Criteria�

Long-Period Flying Qualities Criteria (MIL-F-8785C)!

• Static speed stability"–  No tendency for aperiodic divergence"

•  Phugoid oscillation -> 2 real roots, 1 that is unstable"–  Stable control stick position and force gradients"

•  e.g., Increasing �pull� position and force with decreasing speed"

A.   Non-terminal flight requiring rapid maneuvering"

B.   Non-terminal flight requiring gradual maneuvering"

C.   Terminal flight"

1.   Clearly adequate for the mission"2.   Adequate to accomplish the mission, with

some increase in workload"3.   Aircraft can be controlled safely, but

workload is excessive"

Level of Performance!Flight Phase!


•  Flight path stability [Phase C]"1.   (Δγ/ΔV)SS < 0.06 deg/kt"2.   (Δγ/ΔV)SS < 0.15 deg/kt"3.   (Δγ/ΔV)SS < 0.24 deg/kt"

ΔVSS = aΔδESS + 0( )ΔδTSS + bΔδFSSΔγ SS = cΔδESS + dΔδTSS + eΔδFSS

•  Lecture 19"

Δγ SS

ΔVSS=ca

(with appropriate scaling)

•  From 4th-order model"


• Phugoid stability"1.  Damping ratio ≥ 0.04"2.  Damping ratio ≥ 0"3.  �Time to double�, T2 ≥ 55 sec"

€

T2Ph = −0.693/ζ PhωnPh

Time to Double!

Short Period Criteria"•  Important parameters"

–  Short-period natural frequency"–  Damping ratio"–  Lift slope"–  Step response"

•  Over-/under-shoot"•  Rise time"•  Settling time"•  Pure time delay"

–  Pitch angle response"–  Normal load factor response"–  Flight path angle response (landing)"

Space Shuttle Pitch-Response Criterion"

Short-Period Approximation Transfer Functions"

•  Elevator to pitch rate" Δq(s)ΔδE(s)

=kq s− zq( )


2 ≡kq s+ 1

Tθ2(

)*

+

,-


2

•  Pure gain or phase change in feedback control cannot produce instability"

Bode Plot!

Nichols Chart!

Root Locus!

Short-Period Approximation Transfer Functions"

•  Elevator to pitch angle"•  Integral of prior example"

Δθ(s)ΔδE(s)

=kq s− zq( )

s s2 +2ζSPωnSPs+ωnSP

2( )

•  Pure gain or phase change in feedback control cannot produce instability"

Bode Plot!

Nichols Chart!

Root Locus!

Normal Load Factor"

•  Therefore, with negligible LδE (aft tail/canard effect)"

Δnz =

VNg

Δ α − Δq( ) = −VNg

LαVN

Δα +LδEVN

ΔδE%

&'(

)*

∂Δnz (s)∂ΔδE(s)

=1gLα

∂Δα(s)∂ΔδE(s)

+ LδE%

&'

(

)* ≈

Lαg

%

&'

(

)*∂Δα(s)∂ΔδE(s)

positive down!

positive up!

Δα(s)ΔδE(s)

≈kα


2

•  Elevator to angle of attack (LδE = 0)"

http://www.youtube.com/watch?v=xFemVFgsJAw!

Control Anticipation

Parameter, CAP"•  Inner ear senses angular acceleration about 3 axes"

€

Δ ˙ q (0) = MδE −Mα

VN + Lα

LδE

&

' (

)

* + ΔδESS

ΔnSS =VN

gΔqSS = −

VN

g&

' (

)

* +

MδELα

VN− Mα

LδEVN

& ' ( )

* +

MqLα

VN+ Mα

& ' ( )

* +

ΔδESS

CAP = Δ q(0)

ΔnSS=

− MδE −Mα

VN + LαLδE

%

&'

(

)* Mq

LαVN

+Mα( )LαMδE − LδEMα( ) g

•  Inner ear cue should aid pilot in anticipating commanded normal acceleration"

Initial Angular Acceleration!

Desired Normal Load Factor!

Control Anticipation Factor!

MIL-F-8785C Short-Period Flying Qualities Criterion"

CAP =− Mq

LαVN

+ Mα( )Lα g

≈ωnSP2

nz /α

€

ωnSPvs. nzα

1.   Clearly adequate for the mission"2.   Adequate to accomplish the

mission, with some increase in workload"

3.   Aircraft can be controlled safely, but workload is excessive"

Level of Performance!

with LδE = 0!

• CAP = constant along Level Boundaries"

CAP!Control Anticipation Parameter vs.

Short-Period Damping Ratio "(MIL-F-8785C, Category A)"

CAP =− Mq

LαVN

+Mα( )Lα g

≈ωnSP2

nz /α

C* Criterion"

!  Below Vcrossover, Δq is pilot�s primary control objective"!  Above Vcrossover, Δnpilot is the primary control objective"

C* = Δnpilot +Vcrossover

gΔq

= lpilotΔ q+Δncm( )+VcrossovergΔq

= lpilotΔ q+VNg

Δq−Δ α( )$

%&

'

()+Vcrossover

gΔq

Fighter Aircraft: Vcrossover ≈ 125 m / s

•  Hypothesis"–  C* blends normal load factor at pilot�s location and pitch rate"–  Step response of C* should lie within acceptable envelope"

Gibson Dropback Criterion for Pitch Angle Control"

•  Step response of pitch rate should have overshoot for satisfactory pitch and flight path angle response!

Δq(s)ΔδE(s)

=

kq s+ 1Tθ2

$

%&&

'

())


2

=kq s+ωnSP

ζSP

$

%&

'

()


2

zq −

1Tθ2

= −ωnSP

ζSP

%

&'

(

)*

•  Criterion is satisfied when!

Gibson, 1997!

Lateral-Directional Criteria �

Lateral-Directional Flying Qualities Parameters"

• Lateral Control Divergence Parameter, LCDP!•  ϕ/β Effect"•  ωϕ/ω� Effect!

Lateral Control Divergence Parameter (LCDP)"

•  Aileron deflection produces yawing as well as rolling moment"–  �Favorable yaw� aids the turn command"–  �Adverse yaw� opposes it"

•  Equilibrium response to constant aileron input "

ΔφSΔδAS

=Nβ +Nr

YβVN

%

&'

(

)*LδA − Lβ + Lr

YβVN

%

&'

(

)*NδA

gVN

LβNr − LrNβ( )

•  Large-enough NδA effect can reverse the sign of the response"–  Can occur at high angle of attack "–  Can cause �departure from controlled flight�"

•  Lateral Control Divergence Parameter provides simplified criterion"

LCDP ≡Cnβ−CnδA

ClδA

ClβNβ( )LδA − Lβ( )NδA

LδA= Nβ −

NδA

LδALβ

ωΦ/ωd Effect!

• Aileron-to-roll-angle transfer function "

Δφ(s)ΔδA(s)

=kφ s2 +2ζφωφs+ωφ

2( )s−λS( ) s−λR( ) s2 +2ζDRωnDR

s+ωnDR2( )

�  ωϕ is the �natural frequency� of the complex zeros"�  ωd = ωnDR is the natural frequency of the Dutch roll mode"

• Conditional instability may occur with closed-loop control of roll angle, even with a perfect pilot"

ωϕ/ω� Effect"

• As feedback gain increases, Dutch roll roots go to numerator zeros "•  If zeros are over poles, conditional instability results"

Δφ(s)ΔδA(s)

=kφ s2 +2ζφωφs+ωφ

2( )s−λS( ) s−λR( ) s2 +2ζDRωnDR

s+ωnDR2( )

ϕ/β Effect"

• ϕ/β measures the degree of rolling response in the Dutch roll mode"–  Large ϕ/β: Dutch roll is primarily a rolling motion"–  Small ϕ/β: Dutch roll is primarily a yawing motion"

• Eigenvectors, ei, indicate the degree of participation of the state component in the ith mode of motion"

det sI − F( ) = s − λ1( ) s − λ2( )... s − λn( )λiI − F( )ei = 0

Eigenvectors!• Eigenvectors, ei, are solutions to the equation"

λiI − F( )ei = 0, i = 1,norλiei = Fei , i = 1,n

• For each eigenvalue, the corresponding eigenvector can be found (within an arbitrary constant) from"

Adj λiI − F( ) = a1ei a2ei … anei( ), i = 1,n

MATLABV,D( ) = eig F( )V: Modal Matrix (i.e., Matrix of Eigenvectors)D: Diagonal Matrix of Corresponding Eigenvalues

ϕ/β Effect"• With λi chosen as a complex root of the Dutch roll mode,

the corresponding eigenvector is"

eDR+ =

ereβepeφ

#

$

%%%%%

&

'

(((((DR+

=

σ + jω( )rσ + jω( )βσ + jω( )pσ + jω( )φ

#

$

%%%%%%

&

'

((((((DR+

=

AR ejφ( )rAR e jφ( )

β

AR ejφ( )pAR e jφ( )

φ

#

$

%%%%%%%

&

'

(((((((DR+

• ϕ/β is the magnitude of the ratio of the ϕ and β eigenvectors"

φβ =

AR( )φAR( )β

=VNg

#

$%

&

'( ζDRωnDR

+YβVN

+LβLr

#

$%

&

'(

2

+ ωnDR1−ζDR

2( ),

-..

/

011

12

ϕ/β Effect for the Business Jet Example!

eDR+ =

ereβ

ep

eφ

#

$

%%%%%%%

&

'

(((((((DR+

=

0.5250.4160.6030.433

#

$

%%%%

&

'

((((DR+

φβ = 1.04

Roll/Sideslip Angle ratio in the Dutch roll mode!

Early Lateral-Directional Flying Qualities Criteria!

T12= 0.693 /ζ ωn

v = VNβ

O�Hara, via Etkin!Ashkenas, via Etkin!

Time to Half!

Criteria for Lateral-Directional Modes (MIL-F-8785C)"

Maximum Roll-Mode Time Constant"

Minimum Spiral-Mode Time to Double"

Minimum Dutch Roll Natural Frequency and Damping (MIL-F-8785C)"

Pilot-Vehicle Interactions �

Pilot-Induced Roll Oscillation"

Δφ(s)ΔδA(s) pilot in loop

=Kp /Tp

s +1 /Tp

$

%&

'

()

kφ s2 + 2ζφωφs +ωφ2( )

s − λS( ) s − λR( ) s2 + 2ζDRωnDRs +ωnDR

2( ).

/00

1

233

Aileron-to-Roll Angle Root Locus! Pilot-Aircraft Nichols Chart!

Pilot Transfer Function! Aircraft Transfer Function!YF-16!

YF-17 Landing Approach Simulation"

•  Original design"–  Low short-period natural

frequency"–  Overdamped short period"–  Rapid roll-off of phase angle"–  PIO tendency, CHR = 10"

Elevator-to-pitch angle Nichols chart (gain vs. phase angle)"

80° Phase Margin!

Gibson, 1997!

•  Revised DFCS design"–  Higher short-period natural

frequency"–  Lower short-period damping"–  Reduced time delay in DFCS"–  CHR = 2" Input frequency,

rad/s"

13 dB Gain

Margin!

•  Alternative pilot transfer function: gain plus pure time delay"

H jω( )pilot = KPe− jωτ

•  Gain = constant"•  Phase angle linear in frequency"•  As input frequency increases, ϕ(ω)

eventually > –180°!

But Stability Margins Were Large … How Could CHR = 10?"

KPe− jωτ = KP

KPe− jωτ( ) = − jωτ

H s( )pilot =Δu s( )Δε s( )

= KPe−τ s

Inverse Problem of Lateral Control!

•  Given a flight path, what is the control history that generates it? "–  Necessary piloting

actions "–  Control-law design"

•  Aileron-rudder interconnect (ARI) simplifies pilot input"

Grumman F-14 Tomcat!

Yaw Angle" Roll Angle" Lateral-Stick Command"

Angle of attack (α) = 10 deg; ARI off"

α = 30 deg; ARI off"

α = 30 deg; ARI on"

Stengel, Broussard, 1978!

Flight Control System Design�

Control System Design Methods!•  Linear-quadratic (LQ) regulator"•  Pole placement"•  Parametric optimization"•  Nonlinear inverse dynamics"•  Neural networks"

•  Noisy, incomplete measurements"–  State observer"–  Kalman filter (optimal estimator)"

•  Assume Gaussian errors"•  Combine with LQ regulator"•  LQG regulator"

•  Control at all points in flight envelope"–  Robustness"–  Gain scheduling"–  Adaptive control"

Proportional Stability Augmentation with Command Input!

Δu t( ) = CFΔyC t( ) −CBΔx t( )

Section 4.7, Flight Dynamics"

dim Δu t( )"# $% = m ×1; dim Δx t( )"# $% = n ×1

dim ΔyC t( )"# $% = r ×1, r ≤ m

dim CF[ ] = m × r; dim CB[ ] = m × n

•  Full state feedback"•  Command = desired output"

–  r (≤ m) components"–  Cannot have more independent command inputs,

ΔyC(t), than independent control inputs, Δu(t)"

Proportional Stability Augmentation with Command Input!

Δx t( ) = FΔx t( ) +GΔu t( )Δy t( ) = HxΔx t( ); Hu 0Δu t( ) = CFΔyC t( ) −CBΔx t( )

Δx t( ) = FΔx t( ) +G CFΔyC t( ) −CBΔx t( )#$ %&

= F −GCBΔx t( )#$ %&Δx t( ) +GCFΔyC t( )= FCLΔx t( ) +GCLΔyC t( )

•  Satisfy flying qualities criteria by adjusting gains of the closed-loop command/stability augmentation system"


Dynamics and Control! Substitute Control in Dynamic Equation!

•  Eigenvalues"•  Root loci"•  Transfer functions"•  Bode plots"•  Nichols charts"•  ..."

Next Time:�Maneuvering and Aeroelasticity �

�Reading�

Flight Dynamics, 681-785 �Virtual Textbook, Part 21�

Supplemental Material

Large Aircraft Flying Qualities"•  High wing loading, W/S"•  Distance from pilot to rotational center"•  Slosh susceptibility of large tanks"•  High wing span -> short relative tail length"

–  Higher trim drag"–  Increased yaw due to roll, need for rudder

coordination"–  Reduced rudder effect"

•  Altitude response during approach"–  Increased non-minimum-phase delay in

response to elevator"–  Potential improvement from canard"

•  Longitudinal dynamics"–  Phugoid/short-period resonance"

•  Rolling response (e.g., time to bank)"•  Reduced static stability"•  Off-axis passenger comfort in BWB turns"

Criteria for Oscillations and Excursions (MIL-F-8785C)"

Criteria for Oscillations and Excursions (MIL-F-8785C)!

Proportional-Integral Command and Stability Augmentation!

Δu t( ) = CFΔyC t( ) −CI Δy t( ) − ΔyC t( )#$ %&dt∫ −CBΔx t( )


•  Full state feedback"•  Command = desired output"

–  r (≤ m) components"•  Integral compensation eliminates long-term (bias) errors"


Δx t( ) = FΔx t( ) +GΔu t( )Δy t( ) = HxΔx t( ); Hu 0

Δu t( ) = CFΔyC t( ) −CI Δy t( ) − ΔyC t( )#$ %&dt∫ −CBΔx t( )

Δx t( ) = FΔx t( ) +G CFΔyC t( ) +CI Δy t( ) − ΔyC t( )#$ %&dt∫ −CBΔx t( ){ }= F −GCB[ ]Δx t( ) +G CFΔyC t( ) +CI ΔyC t( ) −HxΔx t( )#$ %&dt∫{ }


Dynamics and Control!

Substitute Control in Dynamic Equation!


Δy t( ) = HxΔx t( ); Hu 0

Δξ t( ) ΔyC t( ) − Δy t( )$% &'dt∫= ΔyC t( ) −HxΔx t( )$% &'dt∫

Δξ t( ) ΔyC t( ) −HxΔx t( )

Δx t( ) = FCLΔx t( ) +GCFΔyC t( ) +GCIΔξ t( )

Δx t( )Δξ t( )

#

$

%%

&

'

((=

FCL GCI−Hx 0

#

$%%

&

'((

Δx t( )Δξ t( )

#

$

%%

&

'

((+

GCL

I

#

$%%

&

'((ΔyC t( )


•  Define integral state, Δξ(t)"•  dim[Δξ(t)] = dim[Δy(t)]"

•  Augmented dynamic equation"

Proportional-Integral Command and Stability

Augmentation!

•  Satisfy flying qualities criteria by adjusting gains of the closed-loop command/stability augmentation system"

•  New modes of motion in augmented system"

•  Eigenvalues"•  Root loci"•  Transfer functions"•  Bode plots"•  Nichols charts"•  ..."

Δχ t( ) Δx t( )Δξ t( )

$

%

&&

'

(

)); dim Δχ t( )$% '( = n + r( ) ×1

Δ χ t( ) = FCL 'Δχ t( ) +GCL 'ΔyC t( )

•  Define augmented state vector"

•  Standard form dynamic equation"

Proportional-Filter Stability Augmentation with Command Input!

Δu t( ) = + CFΔyC t( ) −CBΔx t( ) − −CIΔu t( )#$ %&dt∫


Flight Testing Videos!http://www.youtube.com/watch?v=GXdJxjvQZW4!

http://www.youtube.com/watch?v=t6DdlPoPOE4!

http://www.youtube.com/watch?v=j85jlc1Zfk4!

Atmospheric Hazards to FlightRobert Stengel,

Aircraft Flight Dynamics, MAE 331, 2012"

!  Microbursts"!  Wind Rotors"!  Wake Vortices"!  Clear Air

Turbulence"



Frames of Reference"

!  Inertial Frames"!  Earth-Relative"!  Wind-Relative (Constant Wind)"

!  Non-Inertial Frames"!  Body-Relative"!  Wind-Relative (Varying Wind)"

Earth-Relative Velocity Wind Velocity

Air-Relative Velocity

Angle of Attack, α

Flight Path Angle, γ

Pitch Angle, θ

Pitch Angle and Normal Velocity Frequency Response to Axial Wind"

−

Δθ jω( )ΔVwind jω( )

VNΔα jω( )ΔVwind jω( )

MacRuer, Ashkenas, and Graham, 1973!

!  Pitch angle resonance at phugoid natural frequency"!  Normal velocity (~ angle of attack) resonance at phugoid and

short period natural frequencies"

Pitch Angle and Normal Velocity Frequency Response to Vertical Wind"


!  Pitch angle resonance at phugoid and short period natural frequencies"!  Normal velocity (~ angle of attack) resonance at short period natural frequency"

Δθ jω( )VNΔαwind jω( )

=Δα jω( )

Δαwind jω( )

Sideslip and Roll Angle Frequency Response to Vortical Wind"

=Δβ jω( )Δpwind jω( )

=Δφ jω( )Δpwind jω( )


!  Sideslip angle resonance at Dutch roll natural frequency"!  Roll angle is integral of vortical wind input"

Sideslip and Roll Angle Frequency Response to Side Wind"


!  Sideslip and roll angle resonance at Dutch roll natural frequency"

Δβ jω( )Δβwind jω( )

=Δφ jω( )

Δβwind jω( )

Microbursts"

1/2-3-km-wide �Jet�impinges on surface"

High-speed outflow from jet core"

Ring vortex forms in outlow"

Outflow strong enough to knock down trees"

The Insidious Nature of Microburst Encounter"

!  The wavelength of the phugoid mode and the disturbance input are comparable"

DELTA 191 (Lockheed L-1011)!http://www.youtube.com/watch?v=BxxxevZ0IbQ&NR=1!

Headwind!Downdraft!

Tailwind!Landing Approach!

Importance of Proper Response to Microburst Encounter"

!  Stormy evening July 2, 1994"!  USAir Flight 1016, Douglas DC-9, Charlotte"!  Windshear alert issued as 1016 began descent along glideslope"!  DC-9 encountered 61-kt windshear, executed missed approach"!  Plane continued to descend, striking trees and telephone poles

before impact"!  Go-around procedure was begun correctly -- aircraft's nose rotated

up -- but power was not advanced"!  That, together with increasing tailwind, caused the aircraft to stall "!  Crew lowered nose to eliminate stall, but descent rate increased,

causing ground impact"

Optimal Flight Path Through Worst JAWS Profile"

!  Graduate research of Mark Psiaki"!  Joint Aviation Weather Study (JAWS)

measurements of microbursts (Colorado High Plains, 1983)"

!  Negligible deviation from intended path using available controllability"

!  Aircraft has sufficient performance margins to stay on the flight path"

Downdraft!

Headwind!

Airspeed!

Angle of Attack!

Pitch Angle!

Throttle Setting!

Optimal and 15° Pitch Angle Recovery during

Microburst Encounter"

!  Graduate Research of Sandeep Mulgund!!  Airspeed vs. Time!!  Altitude vs. Time!

!  Angle of Attack vs. Time!

Encountering outflow!

Rapid arrest of descent!

!  FAA Windshear Training Aid, 1987, addresses proper operating procedures for suspected windshear "

Wind Rotors"

Aircraft Encounters with a Wind Rotor"

Radius, ft

TangentialVelocity,

ft/s

!  Tangential velocity vs. radius for Lamb-Oseen Vortex"

Geometry and Flight Condition of Jet Transport Encounters with Wind Rotor"

!  Graduate research of Darin Spilman"

!  Flight Condition"!  True Airspeed = 160 kt"!  Altitude = 1000 ft AGL"!  Flight Path Angle = -3˚"!  Weight = 76,000 lb""!  Flaps = 30˚"!  Open-Loop Control"

!  Wind Rotor"!  Maximum Tangential

Velocity = 125 ft/s"!  Core Radius = 200 ft"

a) co-axial, ψ = 0

b) ψ ≠ 0

3535

vortex

vortex

wind

ψ

35

Typical Flight Paths in Wind Rotor Encounter "

!  from Spilman"

-50

0

50

100

150

200

250

φ [d

eg]

0 5 10 15 20 25 30Time [sec]

ψ i = 0ψ i = 60˚

-60

-50

-40

-30

-20

-10

0

θ [d

eg]

10

20

0 5 10 15 20Time [sec]

25 30

ψ i = 0˚ψ i = 60˚

-300

-200

-100

0

100

200

300

-400 -300

h [f

t]

-200 -100 0 100 200 300y [ft]

400

rotor core radius

flight path

initial entry point

-500

0

500

1000

1500

0 5

h [f

t]

10 15 20 25 30

ψi = 0˚ψi = 60˚

Time [sec]

ground

Linear-Quadratic/Proportional-Integral Filter (LQ/PIF) Regulator"

LQ/PIF Regulation of Wind Rotor Encounter"

!  from Spilman"

-50

0

50

100

150

200

φ [

deg

]

0 2 4 6 8 10Time [sec]

LQR-PIF controlno control input

0

200

400

600

800

1000

1200

h (

AG

L)

[ft]

0 2 4 6 8 10

LQR-PIF control

Time [sec]

no control input

Wake Vortices"

C-5A Wing Tip Vortex Flight Test"http://www.dfrc.nasa.gov/gallery/movie/C-5A/480x/EM-0085-01.mov"

L-1011 Wing Tip Vortex Flight Test"http://www.dfrc.nasa.gov/gallery/movie/L-1011/480x/EM-0085-01.mov"

Models of Single and Dual Wake Vortices"

TangentialVelocity,

ft/s

Radius, ft

Wake Vortex

Wind Rotor

TangentialVelocity,

ft/s

Radius, ft

Wake Vortex Descent and Downwash"

Wake Vortex Descent and Effect of Crosswind"

!  from FAA Wake Turbulence Training Aid, 1995!

Magnitude and Decay of B-757 Wake Vortex "

!  from Richard Page et al, FAA Technical Center"

NTSB Simulation of US Air 427 and FAA Wake Vortex Flight Test"

!  B-737 behind B-727 in FAA flight test"!  Control actions subsequent to wake vortex

encounter may be problematical"!  US427 rudder known to be hard-over from DFDR "

Causes of Clear Air Turbulence"

!  from Bedard"

DC-10 Encounter with Vortex-Induced Clear Air Turbulence"

!  from Parks, Bach, Wingrove, and Mehta!

DC-8 and B-52H Encounters with Clear Air Turbulence"

!  DC-8: One engine and 12 ft of wing missing after CAT encounter over Rockies"

!  B-52 specially instrumented for air turbulence research after some operational B-52s were lost"

!  Vertical tail lost after a severe and sustained burst (+5 sec) of clear air turbulence violently buffeted the aircraft"

!  The Boeing test crew flew aircraft to Blytheville AFB, Arkansas and landed safely"

Conclusions"

!  Critical role of decision-making, alerting, and intelligence"

!  Reliance on human factors and counter-intuitive strategies"

!  Need to review certification procedures"!  Opportunity to reduce hazard through flight

control system design"!  Disturbance rejection"!  Failure Accommodation"

!  Importance of Eternal vigilance"

Supplemental Material�

Alternative Reference Framesfor Translational Dynamics"

!  Earth-relative velocity in earth-fixed polar coordinates:" vE =

VEγ

ξ

#

$

%%%

&

'

(((

vE =VEβEαE

#

$

%%%

&

'

(((

!  Earth-relative velocity in aircraft-fixed polar coordinates (zero wind):"

!  Body-frame air-mass-relative velocity:"

!  Airspeed, sideslip angle, angle of attack"

vA =

u − uw( )v − vw( )w − ww( )

"

#

$$$$

%

&

''''

=

uAvAwA

"

#

$$$

%

&

'''

VAβA

αA

#

$

%%%

&

'

(((=

uA2 + vA

2 + wA2

sin−1 vA /VA( )tan−1 wA /VA( )

#

$

%%%%

&

'

((((

rI = HBI vB

vB =

1mFB vA( ) +H I

Bg I − ω BvB

Θ = LB

I ω B

ω B = IB

−1 MB vA( ) − ω BIBω B#$ %&

!  Rate of change of Translational Position "

!  Rate of change of Angular Position "

!  Rate of change of Translational Velocity "

!  Rate of change of Angular Velocity "

Rigid-Body Equations of Motion"

!  Aerodynamic forces and moments depend on air-relative velocity vector, not the earth-relative velocity vector"

Angle of Attack, α

Flight Path Angle, γ

Pitch Angle, θ

Wind Shear Distributions Exert Moments on Aircraft Through Damping Derivatives"

!  3-dimensional wind field changes in space and time "

wE x,t( ) =wx x, y, z,t( )wy x, y, z,t( )wz x, y, z,t( )

!

"

####

$

%

&&&&E

!  Gradient of wind produces different relative airspeeds over the surface of an aircraft "

!  Wind gradient expressed in body axes " ΔClshear

≈ Clpwing

∂w∂y

− Clpfin

∂v∂x

ΔCmshear≈ Cmqwing ,body ,stab

∂w∂x

ΔCnshear≈ Cnrfin ,body

∂v∂x

WB = HEBWEHB

E

WE =

∂wx ∂x ∂wx ∂y ∂wx ∂z∂wy ∂x ∂wy ∂y ∂wy ∂z

∂wz ∂x ∂wz ∂y ∂wz ∂z

"

#

$$$$

%

&

''''

Aircraft Modes of Motion"

!  Longitudinal Motions"

!  Lateral-Directional Motions"€

ΔLon (s) = s2 + 2ζωns+ωn2( )Ph s

2 + 2ζωns+ωn2( )SP

€

ΔLD (s) = s− λS( ) s− λR( ) s2 + 2ζωns+ωn2( )DR

!  Wind inputs that resonate with modes of motion are especially hazardous"

Natural frequency : ωn , rad / s

Natural Period : Tn =2πωn

, sec

Natural Wavelength : Ln = VN Tp , m

Nonlinear-Inverse-Dynamic Control"

!  Nonlinear system with additive control:"

!  Output vector:"

!  Differentiate output until control appears in each element of the derivative output:"

!  Inverting control law:"

x t( ) = f x t( )!" #$ +G x t( )!" #$u t( )

y t( ) = h x t( )!" #$

yd( ) t( ) = f * x t( )!" #$ +G * x t( )!" #$u t( ) v t( )

u t( ) = G * x t( )!" #$ vcommand − f * x t( )!" #$!" #$

Landing Abort using Nonlinear- Inverse-Dynamic Control"

!  from Mulgund"

400

500

600

700

800

900

1 103

-7500 -5000 -2500 0 2500 5000 7500

Umax

= 60 ft/sec

Umax

= 70 ft/sec

Umax

= 80 ft/sec

Umax

= 90 ft/sec

Umax

= 100 ft/sec

Umax

= 110 ft/sec

Altitude (

ft)

Range (ft)

100

150

200

250

300

350

-7500 -5000 -2500 0 2500 5000 7500

Umax

= 60 ft/sec

Umax

= 70 ft/sec

Umax

= 80 ft/sec

Umax

= 90 ft/sec

Umax

= 100 ft/sec

Umax

= 110 ft/sec

Airs

peed

(ft/s

ec)

Range (ft)

-5

0

5

10

15

-7500 -5000 -2500 0 2500 5000 7500

Umax = 60 ft/sec

Umax = 70 ft/sec

Umax = 80 ft/sec

Umax = 90 ft/sec

Umax = 100 ft/sec

Umax = 110 ft/sec

Alp

ha (d

eg)

Range (ft)

Angle of Attack Limit

Wind Shear Safety Advisor"

!  Graduate research of Alexander Stratton!!  LISP-based expert system"

ON-BOARD DATA

Reactive sensorsWeather radarForward-lookingLightning sensorsFuture products

AIRCRAFT

SYSTEMSCREW

Interface ADVISORY

SYSTEM LOGIC

GROUND-BASEDDATA

LLWASTDWRPIREPSForecastsWeather dataFuture products

Estimating the Probability of Hazardous Microburst Encounter "

!  Bayesian Belief Network"!  Infer probability of hazardous

encounter from "•  pilot/control tower

reports "•  measurements"•  location"•  time of day"

GeographicalLocation

Surface Humidity

Time of Day

ConvectiveWeather

Probability ofMicroburst Wind Shear

Lightning

LightningDetection

Mod/HeavyTurbulence

TurbulenceDetection

Precipitation

WeatherRadar

PilotReport

Low-LevelWind Shear Advisory

SystemAirborne

Forward-LookingDoppler Radar

Reactive Wind ShearAlert System Terminal Doppler

Weather Radar

NTSB Simulation of American 587 "

!  Flight simulation obtained from digital flight data recorder (DFDR) tape"

Digital Flight Data Recorder Data for American 587 "

American 587 (Airbus A-300-600) Encounter with B-747 Wake"

!  PIO and/or aggressive use of rudder seen as possible cause"

!  Aviation Daily, 5/22/02 "!  Boeing Issues Detailed

Guidance On Rudder Use For Roll Control"

Aircraft as Wake Vortex Generators and Receivers"

!  Vorticity, Γ, generated by lift in 1-g flight"

Γ =KgeneratorWρVNb

!  Rolling acceleration response to vortex aligned with the aircraft's longitudinal axis"

Kgenerator

4π

Kreceiver

CLα

2πVNb

p =

Kreceiver12ρVN

2Sb

IxxΓ

Rolling Response vs. Vortex-Generating Strength for 125 Aircraft"

!  Undergraduate summer project of James Nichols"

0.0001

0.001

0.01

0.1

1 10 100

RollingResponse

Vortex Generating Strength

Configuration and Power Effects on Flight Stability


•  Wing design"•  Empennage design"•  Aerodynamic coefficient

estimation and measurement"•  Power Effects"



Loss of Engine"•  Loss of engine produces large yawing

(and sometimes rolling) moment(s), requiring major application of controls "

•  Engine-out training can be as hazardous, especially during takeoff, for both propeller and jet aircraft"

•  Acute problem for general-aviation pilots graduating from single-engine aircraft"

Beechcraft Baron!Learjet 60!

Solutions to the Engine-Out Problem"

•  Engines on the centerline (Cessna 337 Skymaster)"

•  More engines (B-36)"•  Cross-shafting of engines (V-22)"•  Large vertical tail (Boeing 737)"

NASA TCV (Boeing 737)!

Cessna 337!

Convair B-36!

Boeing/Bell V-22!

Airplane Balance"•  Conventional aft-tail configuration "

–  c.m. near wing's aerodynamic center (point at which wing's pitching moment coefficient is invariant with angle of attack ~25% mac)"

•  Tailless airplane: c.m. ahead of the neutral point"

Douglas DC-3!

Northrop N-9M!

Airplane Balance"•  Canard configuration: "

–  Neutral point moved forward by canard surfaces"–  Center of mass may be behind the neutral point, requiring

closed-loop stabilization"•  Fly-by-wire feedback control can expand envelope

of allowable center-of-mass locations (e.g., open-loop instability"

Grumman X-29!

McDonnell-Douglas X-36!

Configuration Effects Can Be Evaluated via Approximate Dynamic Models"

λRoll ≈ Lp ≈ Clp̂

ρVN4Ixx

$

%&'

()Sb2

ωn ≈ − Mα + MqLαVN

%

&'(

)*; ζ ≈

LαVN

− Mq%

&'(

)*

2 − Mα + MqLαVN

%

&'(

)*

ωnDR≈ Nβ 1−

YrVN( ) + Nr

YβVN

ζDR ≈ − Nr +YβVN

&'(

)*+

2 Nβ 1−YrVN( ) + Nr

YβVN

•  Phugoid Mode"

•  Short-Period Mode"

•  Dutch Roll Mode"

•  Roll Mode"

ωn ≈ gLV /VN ; ζ ≈ DV

2 gLV /VN

λSpiral ≈ 0•  Spiral Mode"However, important mode-coupling terms, e.g., MV and L!,

are neglected "

•  Straight Wing"–  Subsonic center of

pressure (c.p.) at ~1/4 mean aerodynamic chord (m.a.c.)"

–  Transonic-supersonic c.p. at ~1/2 m.a.c."

•  Delta Wing"–  Subsonic-supersonic

c.p. at ~2/3 m.a.c."

Planform Effect on Center of Pressure Variation with Mach Number"

•  Mach number "–  increases the static margin of conventional configurations -> Short Period"–  Has less effect on delta wing static margin"

Cmα

Sweep Reduces Subsonic Lift Slope"

CLα=

πAR

1+ 1+ AR2cosΛ1 4

$

%&

'

()

2

1− M 2 cosΛ1 4( )+

,

---

.

/

000

=πAR

1+ 1+ AR2cosΛ1 4

$

%&

'

()

2+

,

---

.

/

000

[Incompressible flow]

CLα=

2π 2 cotΛLE

π + λ( )where λ = m 0.38 + 2.26m − 0.86m2( )

m = cotΛLE cotσΛLE , σ : measured from y axis

Swept Wing"

Triangular Wing"

Effects of Wing Aspect Ratio"•  Wing lift slope has direct effect on"

–  Phugoid damping"–  Short period natural frequency and damping"–  Roll damping"

λRoll ≈ Lp ≈ Clp̂

ρVN4Ixx

$

%&'

()Sb2

ωn = − Mα + MqLαVN

$

%&'

(); ζ =

LαVN

− Mq$

%&'

()

2 − Mα + MqLαVN

$

%&'

()

ωn ≈ 2 gVN; ζ ≈

12 L / D( )N

Short Period"

Phugoid"

Roll"

Effects of Wing Aspect Ratio and Sweep Angle"

•  Lift slope"•  Pitching moment slope"•  Lift-to-drag ratio"•  All contribute to"

–  Phugoid damping"–  Short period natural frequency and

damping"–  Roll damping"

CLα,Cmα

,Clp,Clβ

•  !c/4 = sweep angle of quarter-chord"

•  Sweep moves lift distribution toward wing tips"–  Roll damping"–  Static margin"

•  Sweep increases dihedral effect of wing!

CLα,Cmα

,Clp,Clβ

Sweep Effect on Lift Distribution" Wing Location and Angle Effects"

•  Vertical location of the wing, dihedral angle, and sweep"–  Sideslip induces yawing motion"–  Unequal lift on left and right

wings induces rolling motion"•  Lateral-directional (spiral mode)

stability effect (TBD)" Clβ

Modes Strongly Affected By The Empennage"

ωn ≈ − Mα + MqLαVN

%

&'(

)*

ζ ≈LαVN

− Mq%

&'(

)*2 − Mα + Mq

LαVN

%

&'(

)*

ωnDR≈ Nβ + Nr

YβVN

ζDR ≈ − Nr +YβVN

&'(

)*+

2 Nβ + NrYβVN

•  Short-Period Mode (horizontal tail)"

•  Dutch Roll Mode (vertical tail)"

•  �Weathervane� and damping effects"

Cmα

,Cmq,Cm α

,Cnβ,Cnr

,Cn β

•  Increased tail area with no increase in vertical height"•  End-plate effect for horizontal tail improves effectiveness"•  Proximity to propeller slipstream"

Twin and Triple Vertical Tails"

North American B-25!

Lockheed C-69!

Consolidated B-24!

Fairchild-Republic A-10!

•  Increase directional stability"•  Counter roll due to sideslip of the dorsal fin

""

LTV F8U-3!

Ventral Fin Effects"

North American X-15!

Learjet 60!Beechcraft 1900D!

Cnβ,Cnr

,Cn β,Clβ

Ground Attack Aircraft"•  Maneuverability, payload, low-speed/subsonic performance, ruggedness"

General Aviation Aircraft"•  Low cost, safety, comfort, ease of handling"

Approaches to Stealth"•  Low radar cross-section"

•  Open-loop instability"•  Need for closed-loop control"

Supersonic Flight"•  Low parasitic drag, high supersonic L/D" Hypersonic Flight"

•  Transient vs. cruising flight"•  Hypersonic performance"•  Resistance to aerodynamic

heating"

Commercial Transport"

•  Safety, fuel economy, cost/passenger-mile, maintenance factors"

•  Regional vs. long-haul flight segments"

Business Aircraft"•  Segment between personal and commercial transport"

Long-Range/-Endurance Surveillance Aircraft"

•  Subsonic performance"

Propeller Effects"

•  Slipstream over wing, tail, and fuselage"–  Increased dynamic pressure"–  Swirl of flow"–  Downwash and sidewash at the tail"

•  DH-2 unstable with engine out"•  Single- and multi-engine effects"•  Design factors: fin offset (correct at one

airspeed only), c.m. offset"•  Propeller fin effect: Visualize lateral/

horizontal projections of the propeller as forward surfaces"

•  Counter-rotating propellers minimize torque and swirl"

Westland Wyvern!

DeHavilland DH-2!

DeHavilland DHC-6!

Cmα

,Cmq,Cm α

,Clo,Cno

,Cnβ,Cnr

,Cn β

Jet Effects on Rigid-Body Motion"•  Normal force at intake (analogous to propeller fin effect) (F-86)"•  Deflection of airflow past tail due to entrainment in exhaust (F/A-18)"•  Pitch and yaw damping due to internal exhaust flow"•  Angular momentum of rotating machinery"

North American F-86! McDonnell Douglas F/A-18!

Cmo

,Cmα,Cmq

,Cno,Cnβ

,Cnr,Cn β

Next Time:�Problems of High Speed

and Altitude��

Reading�Flight Dynamics,�

Aircraft Stability and Control,�Virtual Textbook, Part 22 �

•  Greek mythology: Daedalus and Icarus!–  Attempt to escape from Crete!

•  863 BC: Bladud (King Lear�s father)!–  9th king of England !–  First tower-jumper!–  Wings of feathers!–  Died of a broken neck!

Aviation History: Mythology to 1990




Antiquity�

Experimentalists"•  Henri Pitot: Pitot tube (1732)!

•  Benjamin Robins: Whirling arm "wind tunnel� (1742)!

•  Sketches "modern" airplane configuration (1799)!

•  Hand-launched glider (1804)!•  Papers on applied

aerodynamics (1809-1810)!•  Triplane glider carrying 10-yr-

old boy (1849)!•  Monoplane glider carrying

coachman (1853)!–  Cayley's coachman had a steering

oar with cruciform blades!•  Modern reconstruction (right)!

Sir George Cayley!

Visionaries and Theorists!

•  1831: Thomas Walker!–  Various glider concepts!–  Tandem-wing design influenced Langley!

•  1843: William Henson & John Stringfellow !–  Aerial steam carriage concept!–  Vision of commercial air transportation (with

Marriott and Columbine, The Aerial Transit Company)!

•  1860-1900: Theoretical aerodynamic contributions!

–  Helmholtz!–  Kirchhoff!–  Lord Rayleigh!–  Reynolds!–  Lanchester, and others!

19th Century Flyers"•  1868: Jean Marie Le Bris!

–  Artificial Albatross glides a short distance!•  1891-96: Hang-glider flights!

–  Otto Lilienthal!–  Chanute, Pilcher, ...!

•  1890: Clement Ader!–  Steam-powered Eole hops!

•  1894: Sir Hiram Maxim!–  Steam-powered biplane hops!–  Vertical gyro/servo control of the

elevator!

•  1884: Alexander Mozhaisky's steam-powered manned airplane!

–  brief hop off the ground!–  flat-plate wings!

•  1874: Felix du Temple's hot-air engine manned monoplane!

–  Flies down a ramp!Early 20th Century�

Samuel Pierpoint Langley"

•  Astronomer supported by Smithsonian Institution!•  Whirling-arm experiments!•  1896: Langley's steam-powered Aerodrome model

flies 3/4 mile!•  1903: Manned aircraft flight ends in failure!

The Wright Brothers"

•  Wilbur and Orville were bicycle mechanics from Dayton, OH!

•  Self-taught, empirical approach to flight!•  Wind-tunnel, kite, and glider experiments!

•  1903: Powered, manned aircraft flight ends in success!

Wright Brothers� Technical Contributions"

•  Recognized importance of balance and steering!–  Bank to turn preferable to �skid turn�!

–  Roll control induced yaw!–  Too much stability!

•  hinders control!•  increases response to gusts!

–  Wings can stall!•  Experimental gliders!

–  Wing warp controlled roll!–  Foreplanes controlled pitch!

•  2nd glider!–  Vertical tail !–  Coupled to the wing warp to

suppress adverse yaw!

After Kitty Hawk!•  1906: 2nd successful aviator: Alberto

Santos-Dumont, standing!!–  High dihedral, forward control surface!

•  Wrights secretive about results until 1908; few further technical contributions!

•  1908: Glenn Curtiss et al incorporate ailerons!

–  Wright brothers sue for infringement of 1906 US patent (and win)! •  1909: Louis Bleriot's flight

across the English Channel!

Glenn Curtiss"•  1908: Glenn Curtiss becomes dominant US aviation inventor!•  1914: Langley�s Aerodrome finally flies!

–  Curtiss et al modify and fly Langley Aerodrome!–  Unsuccessful effort to discredit Wright patent!–  Alexander Graham Bell�s support!

Stability vs. Control OR

Stability and Control?!

•  Prior to 1903, it was thought that an airplane should hold its course alone!–  Pilot could steer by deflecting the rudder !

•  This suggested:!–  Aft-mounted tail!–  Wing dihedral or high wing!–  Proper center-of-mass location!

Early Aircraft Control"•  Lillienthal shifted center of mass in

hang gliders!•  Langley's plane had movable

cruciform tail!

•  Wright brothers used wing warping and movable rudder and elevator surfaces!

•  Glenn Curtiss invented aileron surfaces!

Pilot Inputs to Control: the Wright Approach"•  1903 Wright Flyer!

–  Prone pilot !–  Stick for pitch control!–  Hip cradle for wing-warping roll

control!–  Aileron-rudder interconnect (ARI)!–  http://www.airspacemag.com/how-

things-work/1903-flyer.html!•  1905 Wright Flyer !

–  Upright pilot !–  Left lever for pitch!–  Right lever for roll and yaw with ARI!–  Right lever later modified to separate

roll (left-right) and yaw (fore-aft) control) w/o ARI!

–  Feet not used for control!–  http://wright.nasa.gov/airplane/

air1905.html!

Control Linkages: the Bleriot Approach"

•  Louis Bleriot introduced:!–  Rudder bar controlled by feet!–  Center stick for pitch and roll control!

Bleriot XI!

http://en.wikipedia.org/wiki/Blériot_XI! http://ohtm.org/blg/collections/aircraft/1909-bleriot-xi-representation/!

Aviation in The Great War"•  1914-18: World War I changes the

complexion of flying!–  Reconnaissance!–  Air superiority (dog fights) !–  Bombing!–  Personal transport!

•  Wrights� US monopoly broken by licensing for war effort!

•  Aircraft Design!–  Biplanes, a few mono- and

triplanes!–  Design for practical functions!–  Multiple engines, larger aircraft!–  Aft tails!–  Increased maneuverability,

speed, g-loads, altitude!–  Improved piston engines!–  Tractor propellers!

SPAD S.VII!

World War I fighter replicas in flight!http://www.youtube.com/watch?v=JsT_rgScrg0!

Maneuvering World War I Aircraft"

•  Maneuverable aircraft with idiosyncrasies!–  Rotary engine!–  Small tail surfaces!–  Reliability issues!

•  Maneuvering to stalls and spins!•  Snap roll: rudder and elevator!•  Barrel roll : aileron !•  Cross-control (e.g., right rudder, left

stick)!–  glide path control during landing !–  good view of landing point!

•  Unintended snap rolls led to spins and accidents during takeoff or landing!

http://www.youtube.com/watch?v=OBH_Mb0Kj2s!

http://www.youtube.com/watch?v=6ETc1mNNQg8!

Sopwith Camel"

•  Rotary engine induced gyroscopic coupling!•  Highly maneuverable!•  Aft fuel tank; when full, center of mass was too far aft

for stability!•  Vertical tail too small, spin recovery not automatic

with centering of controls!

http://www.youtube.com/watch?v=3ApowyEXSXM!

S.E.-5 vs. Fokker D.VII"•  RAF S.E.-5: theoretical approach to design!

–  �Best WWI design from the Royal Aircraft Factory�!

–  Stationary engine!–  High dihedral!–  Stable spiral mode!–  High control forces!–  Poor maneuverability!–  Relatively safe and effective!

•  Fokker D.VII: empirical approach to design!

–  Horn balances to reduce control forces!–  Stationary engine!–  Neutral-to-negative stability!–  Good maneuverability!–  Relatively dangerous!

The Correct Answer: Stability AND Control"

•  Need for better understanding of Flying (or Handling) Qualities !–  Stability and controllability characteristics as perceived

by the pilot!•  Desired attributes: Stability of the S.E.-5 and

controllability of the D.VII!

http://www.youtube.com/watch?v=kwXcwu6JQk8!

Between the Wars �

Aviation Between the Wars"•  1918-38: !

–  Birth of airlines!–  Trophy races

http://www.youtube.com/watch?v=R3AROX6OO88!

–  Aviation firsts (Lindbergh crosses the Atlantic, 1927) http://www.youtube.com/watch?v=uIUL_qUJUOo&feature=related!

–  Flying boats !–  Sport aviation!

Curtiss R3C-2! Gee-Bee R-1!

Ryan NYP!

http://www.youtube.com/watch?v=DwqYh995YhU!

Air Commerce Act of 1926"•  Airlines formed to carry mail and passengers: !

–  Northwest (1926)!–  Eastern (1927), bankruptcy!–  Pan Am (1927), bankruptcy!–  Boeing Air Transport (1927), became United (1931)!–  Delta (1928), consolidated with Northwest, 2010!–  American (1930)!–  TWA (1930), acquired by American!–  Continental (1934), consolidated with United, 2010!

Boeing 40!

Ford Tri-Motor! Lockheed Vega !

http://www.youtube.com/watch?v=3a8G87qnZz4!

Commercial Aircraft of the 1930s!•  Streamlining, engine cowlings!•  Douglas DC-1, DC-2, DC-3!

•  Lockheed 14 Super Electra, Boeing 247, exterior and interior!

Seaplanes Became the First TransOceanic Air Transports!

•  PanAm led the way!–  1st scheduled TransPacific flights(1935)!–  1st scheduled TransAtlantic flights(1938)!–  1st scheduled non-stop Trans-Atlantic flights (VS-44, 1939)!

•  Boeing B-314, Vought-Sikorsky VS-44, Shorts Solent!•  Superseded by more efficient landplanes (lighter, less drag)!

http://www.youtube.com/watch?v=x8SkeE1h_-A!

1930s Air Racers Presage Fighters of World War II"

•  Aircraft Design !–  Transition to

monoplanes!–  Metal skins and

structure!–  Semi-monocoque design!–  Improved aerodynamics!–  Improved in-line, V, and

radial engines !–  Increased

maneuverability, speed, altitude!

–  Seaplanes faster than landplanes (why?)!

Supermarine S.6B!

Hughes H-1 (replica)!

Macchi MC72!

V = 566 km/hr!

V = 709 km/hr!

V = 547 km/hr!

World War II�

http://www.youtube.com/watch?v=hgyFzUVtDOY&feature=related!

Technology of World War II Aviation"•  1938-45: Analytical and experimental

approach to design!–  Many configurations designed and

flight-tested !–  Increased specialization; radar,

navigation, and communication!–  Approaching the "sonic barrier�!

•  Aircraft Design! !–  Large, powerful, high-flying aircraft!–  Turbocharged engines!–  Oxygen and Pressurization!

Spitfire!

P-51D!

B-17!

http://www.youtube.com/watch?v=stKz-elSYy0!

Power Effects on Stability and Control!•  Gee Bee R1 Racer: an engine with

wings and almost no tail"•  During W.W.II, the size of fighters

remained about the same, but installed horsepower doubled (F4F vs. F8F)!

•  Use of flaps means high power at low speed, increasing relative significance of thrust effects"

•  Short-Takeoff-and-Landing (STOL) aircraft augment takeoff/landing lift in many ways, e.g.,"

–  Full-span flaps"–  Deflected thrust"

Grumman F8F!

Grumman F4F!

GB R1!

CTo,CTV

,CTδT,CLδT

World War II Carrier-Based Airplanes!

•  Takeoff without catapult, relatively low landing speed http://www.youtube.com/watch?v=4dySbhK1vNk!

•  Tailhook and arresting gear!•  Carrier steams into wind!•  Design for storage (short tail length,

folding wings) affects stability and control!

Chance-Vought F4U Corsair!

Grumman TBF!

Douglas TBD!

F4U flight"http://www.youtube.com/watch?v=JANR0XPtVzw!

SBD Dauntless Flight"http://www.youtube.com/watch?v=aiJhcKgg4eE&feature=related!

F4U!

Carrier Crash Landings!http://www.youtube.com/watch?v=H8Bim7-hZfg!

Grumman TBF !

Multi-Engine Aircraft of World War II!

•  Large W.W.II aircraft had unpowered controls:!–  High foot-pedal force!–  Rudder stability problems

arising from balancing to reduce pedal force!

•  Severe engine-out problem for twin-engine aircraft!

Boeing B-17! Boeing B-29!Consolidated B-24!

Douglas A-26!


Martin B-26!

WW II Military Flying Boats!•  Seaplanes proved useful during World War II!

Lockheed PBY Catalina!

Martin PBM Mariner !

Martin PB2M Mars!

Boeing XPBB Sea Ranger!

Saunders-Roe SR.36 Lerwick!Grumman JRF-1 Goose!

Birth of the Jet Airplane"

Bell P-59A (1942)!

Gloster Meteor (1943)!

Messerschitt Me 262 (1942)!

Heinkel He. 178 (1939)!

Gloster E/28/39 (1941)!Early Jet Aircraft�

From Propellers to Jets!Douglas XB-43!

Douglas XB-42 Mixmaster!

Convair B-36!Convair YB-60!

Northrop YB-35! Northrop XB-49!

Jets at an Awkward Age!•  Performance of the first jet aircraft

outstripped stability and control technology!–  Lacked satisfactory actuators,

sensors, and control electronics!–  Transistor: 1947, integrated circuit:

1958!•  Dramatic dynamic variations over

larger flight envelope!–  Control mechanisms designed to

lighten pilot loads were subject to instability!

•  Reluctance of designers to embrace change, fearing decreased reliability, increased cost, and higher weight !


Lockheed P-80!

Douglas F3D!

Convair XF-81!

From Straight to Swept Wings!•  Straight-wing models were redesigned with swept wings to

reduce compressibility effects on drag!•  Dramatic change in stability, control, and flying qualities!•  North American FJ-1

and FJ-4 Fury!•  Republic F-84B Thunderbird

and F-84F Thunderstreak!•  Grumman F9F-2 Panther

and F9F-6 Cougar! Toward Supersonic Flight and Stealth�

Fighter Jets of the 1950s!•  Emphasis on supersonic speed!

Republic F-105!

Lockheed F-104!

Convair F-102!

McDonnell F-101!North American F-100!

Fighter Jets of the 1960s!•  Emphasis on mission!

McDonnell F-4!Dassault Mirage F1!

Convair F-106!Grumman A-6!

Fighter Jets of the 1980s!•  Emphasis on performance!

Repbulic F-105!Lockheed-Martin F-117!

Panavia Tornado!

BAE/McDonnell-Douglas AV-8B!

Mig-31A!

Personal and Business Aircraft�

Early Concepts for Safe Personal Aircraft!•  Low takeoff and landing speeds!•  Benign flying qualities!•  �Stall/spin-proof� designs!•  Ercoupe!

–  Limited control authority!–  Control wheel, ARI, no rudder pedals!–  Limited center-of-mass travel!–  Limited speed range!–  Wing leveling and lateral stability!–  Fixed, tricycle landing gear!

Curtiss Tanager!

Piper J-3 Cub!

Ercoupe!

Less-Safe Personal Airplanes!•  Mignet Flying Flea (Homebuilt, pivoting main wing, no ailerons, unrecoverable dive)!•  V-tail Beechcraft Bonanza Model 35 (10,000 built, 250 in-flight structural failures) !•  American Yankee AA-1 (BD-1, �hot�, stalls and spins) !•  Bede BD-5 (Home-built, unforgiving flying qualities)!

Propeller-Driven Personal Aircraft!•  Single reciprocating engine, mechanical controls, fixed or

retracting gear, high price!•  Cirrus SR-20/22 has a recovery parachute (used 13 times

through 2008, saved 24 lives; 2 parachute failures)!

Piper Malibu!

Cessna 172!

Cirrus SR20/22!Beech Bonanza A36!

Mooney M20!

Business Jets!•  Twin turbojet/fan engines!

Gulfstream II!

Cessna Citation I!

Learjet 24!

North American Sabreliner!

Commercial Transport Aircraft�

Commercial Aircraft of the 1940s!•  Pre-WWII designs!•  Derivatives of military transport and

bomber aircraft of WWII!–  Convair 240 (> C-131, T-29)!–  Douglas DC-4 (> C-54)!–  Boeing Stratoliner 377 (from B-29, C-97)!–  Lockheed Constellation 749 (from C-69)!

Commercial Propeller-Driven Aircraft of the 1950s!•  Introduction of the turboprop engine!•  Douglas DC-6, DC-7, Lockheed Starliner 1649, Vickers Viscount, Bristol

Britannia, Lockheed Electra 188!

Commercial Jets of the 1950s!•  Low-bypass ratio turbojet

engines!•  deHavilland Comet (1954)!

–  1st commercial jet transport!–  engines buried in wings!

•  Boeing 707 (1957)!–  derived from USAF KC-135!–  engines on pylons below wings!–  largest aircraft of its time!

•  Sud-Aviation Caravelle (1959)!–  1st aircraft with twin aft-

mounted engines!

Small Commercial Jets of the 1960s!

•  Preponderance of aft-mounted 2- and 3-engine configurations (BAC 1-11, Douglas DC-9)!

•  Boeing 727 (1963)!–  1st with 3 aft-mounted engines!

•  Hawker-Siddeley Trident (1964)!•  Boeing 737 (1967)!

Large Commercial Jets of the 1960s!•  Pylon-mounted 4-engine configurations

(Convair 880, DC-8, Stretched DC-8, Boeing 747)!•  DC-8 design was well-suited to a stretch

(fuselage plugs fore and aft of wing!•  B-707 was not!

Commercial Jets of the 1970s!•  High-bypass turbofan engines!•  Introduction of the 2/3-engine

jumbo jets!–  Lockheed L-1011!–  Douglas DC-10!–  Airbus A300!

•  Supersonic transports!–  BAC/SA Concorde (1972)!–  Tupolev Tu-144 (1977)!

Lockheed L-1011!

Douglas DC-10!

Airbus A300!

BAC/SA Concorde!

Tupolev Tu-144!

Commercial Jets of the 1990s!•  Derivatives of 1980s designs

(McDonnell-Douglas MD-11, Boeing 747-400, Airbus A330)!

•  First Boeing fly-by-wire design (B-777)!•  First 4-engine Airbus (A340)!

McDonnell-Douglas MD-11!

Boeing 747-400!

Airbus A330!

Boeing 777!

Boeing 247!

Airbus A340!

Daedalus and Icarus Revisited, 23 April 1988 �

•  Human-powered airplane, MIT Daedalus, flies 115 km from Crete to the mainland (Santorini) in 3 hr, 54 min�

•  Empty mass = 31 kg�

http://en.wikipedia.org/wiki/MIT_Daedalus�

http://www.youtube.com/watch?v=l131fSveof8!

Next Time:�Configuration Aerodynamics�

�Reading�

Flight Dynamics, 65-84!Virtual Textbook, Part 4,5 �


Natural Philosophers and Theorists"•  350 BC: Aristotle!

–  Continuum model!–  Suggests that a body

moving through continuum would encounter resistance !

•  250 BC: Archimedes!–  Fluid set in motion by

pressure differential!•  1490: Da Vinci ! !

–  Cross-sectional area times Velocity = constant (continuity)!

–  Sketches of flow patterns!–  Ornithopter and helicopter

concepts!

•  1669-87: Newton!–  Newton's Laws!–  Newtonian flow, sin2 !

force dependency!•  1738: Bernoulli!

–  Pressure-velocity relationship!

•  1752: Euler !–  Equations for fluid flow!

•  1788: Lagrange!–  Velocity potential and

stream function!

The Early Wright Flyers"

•  1903 Wright Flyer was very unstable, almost unmanageable !

•  In 1904-5:!–  Removed wing anhedral

(negative dihedral)!–  Increased rudder and

elevator area!–  Rudder controlled by

separate lever!–  Center of mass moved

forward!

Those Magnificent Men in Their Flying Machines"

•  Aircraft Design!–  Biplanes and monoplanes !–  Thin, cambered wings!–  Fore and aft horizontal tails!–  Aft vertical tails!

–  Wooden frames and struts, wire bracing, canvas covering!

–  Gasoline engines, improved efficiency!

–  Invention of rotary engine!

Stability and Control Analysts"•  Frederick Lanchester (1868-1946) !

–  Model gliders!–  Two books, Aerial Flight and

Aerodynetics, 1907!–  Identified the phugoid mode!–  Mechanical engineer, built

innovative motor cars!–  Operations research!

•  Lanchester�s Power Laws!

•  George H. Bryan (1864-1928)!–  Longitudinal equations of motion

(with W.E. Williams, 1903)!–  Full equations of motion and

linearized equations (1911)!

Comfort and Elegance by the End of the Decade!•  Boeing 307, 1st pressurized cabin (1936), flight engineer, B-17 pre-cursor, large

dorsal fin (exterior and interior)!

•  Sleeping bunks on transcontinental planes (e.g., DC-3)!•  Full-size dining rooms on flying boats!

Early Carrier Jets!•  Problems exacerbated for jet

aircraft!–  slower thrust response!–  higher approach speeds!–  reduced phugoid damping!–  lower lift-slope wings!–  higher angles for trim!

McDonnell FH-1 Banshee!

McDonnell F3H-1 Demon !

North American FJ-2 Fury!

Vought F7U Cutlass!

Grumman F9F Panther !

Jet-Powered Sea Planes!•  Primarily experimental aircraft!•  Overtaken by events!

Convair XF2Y Sea Dart !Saunders-Roe SR.A/1!

Beriev Be-200!

Martin XP6M Seamaster!

•  ... but perhaps the Be-200 (2004) may change that!

Fighter Jets of the 1970s!•  Emphasis on agility and attack!

Northrop F-5E!

Grumman F-14!

McDonnell-Douglas F/A-18!

General Dynamics F-16!

Republic A-10!

McDonnell-Douglas F-15!

Commercial Jets of the 1980s!•  Extensions of the 1970s transports (Boeing

757, 767, Airbus A310, A320)!•  Introduction of fly-by-wire control, side-stick

hand controllers, �glass cockpit� (A320)!•  Cockpit commonality!

Boeing 757!

Boeing 767!

Airbus A310!

Airbus A320!

Development of Commercial Jets Over 52 Years!

Boeing 707-120B Airbus A340-300 Boeing 787-3Date of Service Entry 1958 1991 2011Cockpit crew 3 2 2Passengers 110 (2 class) 335 (2-class, typical) 290-330Length 145 ft 1 in (44.07 m) 63.60 metres (208 ft 8 in) 186 ft (56.7 m)Wingspan 145 ft 9 in (44.42 m) 60.30 metres (197 ft 10 in) 170 ft 6 in (52.0 m)

Wing Sweep Angle 37.5 deg 30 deg 32.2 degWing Area 3,010 square feet (280 m2) 361.6 square metres (3,892

sq ft) 3,501 sq ft (325 m2)Tail height 42 ft 5 in (12.93 m) 16.85 metres (55 ft 3 in) 55 ft 6 in (16.9 m)Fuselage width 12 ft 4 in (3.76 m) 5.64 metres (18 ft 6 in) 18 ft 11 in (5.77 m)Maximum Takeoff Weight (MTOW)

257,000 lb (116,570 kg) 276,500 kilograms (610,000 lb)

375,000 lb (170,000 kg)

Empty weight 122,533 lb (55,580 kg) 130,200 kilograms (287,000 lb)

223,000 lb (101,000 kg)

Runway needed at MTOW 11,000 ft (3,330 m) 2,990 metres (9,810 ft) TBDFuel Capacity 17,330 US gal (65,590 l) 147,850 litres (39,060 US

gal)12,830 US gal (48,567 L)

Range at MTOW (max fuel) 4,700 nmi (8,704 km) 7,400 nautical miles (13,700 km; 8,500 mi)

2,500–3,050 nmi (4,630–5,650 km; 2,880–3,510 mi)

Cruising speed Mach 0.82 Mach 0.82 Mach 0.85Powerplants 17,000 lbf (75.6 kN) x 4 139–151 kilonewtons

(31,000–34,000 lbf) x 453,000 lbf (240 kN) x 2

Aerobatics"•  Flat Spins Upright Spins and Snap Roll Spins!

http://www.youtube.com/watch?v=pdPBiy0niAc!

http://www.youtube.com/watch?v=84A6cDlSKzY!

•  Barrel Roll, Aileron Roll, Loop, Hammerhead!

http://www.youtube.com/watch?v=tOZEgKXJMCE!

•  Bob Hoover in his Aero Commander Shrike!

http://www.youtube.com/watch?v=TRt6UnNzR6I&feature=related!

•  Bob Hoover in his F-86 Sabre!

http://www.youtube.com/watch?v=uRmX3pixfj4!•  Learjet!

Cruising Flight Performance Robert Stengel, Aircraft Flight Dynamics,

MAE 331, 2012!



•  U.S. Standard Atmosphere"•  Airspeed definitions"•  Steady, level flight"•  Simplified power and thrust models"•  Back side of the power/thrust curve"•  Performance parameters"•  Breguet range equation"

–  Jet engine"–  Propeller-driven (Supplement)"

U.S. Standard Atmosphere, 1976"

http://en.wikipedia.org/wiki/U.S._Standard_Atmosphere!

Dynamic Pressure and Mach Number"ρ = airdensity, functionof height= ρsealevele

−βh

a = speed of sound= linear functionof height

Dynamic pressure = q ρV 2 2Mach number =V a

Definitions of Airspeed"

•  Indicated Airspeed (IAS)"

•  Calibrated Airspeed (CAS)*"

•  Airspeed is speed of aircraft measured with respect to air mass"–  Airspeed = Inertial speed if wind speed = 0"

IAS = 2 pstagnation − pambient( ) ρSL =2 ptotal − pstatic( )

ρSL

=2qcρSL

, with qc impact pressure

CAS = IAS corrected for instrument and position errors

=2 qc( )corr−1

ρSL* Kayton & Fried, 1969; NASA TN-D-822, 1961!

Definitions of Airspeed"

•  True Airspeed (TAS)*"

•  Equivalent Airspeed (EAS)*"

•  Airspeed is speed of aircraft measured with respect to air mass"–  Airspeed = Inertial speed if wind speed = 0"

EAS =CAS corrected for compressibility effects =2 qc( )corr−2

ρSL

V TAS = EAS ρSL

ρ(z)= IAScorrected

ρSLρ(z)

•  Mach number"

M =TASa

* Kayton & Fried, 1969; NASA TN-D-822, 1961!

Air Data System"

Air Speed Indicator!Altimeter!Vertical Speed Indicator!

Kayton & Fried, 1969!

•  Subsonic speed: no shock wave ahead of pitot tube"•  Supersonic speed: normal shock wave ahead of pitot tube"

Dynamic and Impact Pressure"

•  Dynamic pressure also can be expressed in terms of Mach number and static (ambient) pressure"

q ρV 2 2 : Dynamic pressureqc = ptotal − pstatic : Impact pressure

pstat z( ) = ρamb z( )RT z( ) [Ideal gas law, R = 287.05 J/kg-°K]

a z( ) = γRT z( ) [Speed of sound, T = absolute temperature, °K, γ = 1.4]

M =V a [Mach number]

q ρamb z( )V 2 2 = γ

2pstat z( )M 2

•  In incompressible flow, dynamic pressure = impact pressure"

Substituting!

•  In subsonic, isentropic compressible flow"

•  Impact pressure is"

ptotal z( )pstatic z( )

= 1+ γ −12

M 2#

$%

&

'(γ γ−1( )

qc ptotal z( )− pstatic z( )"# $%= pstatic z( ) 1+0.2M 2( )3.5 −1"

#&$%'

Compressibility Effects on Impact Pressure"

•  In supersonic, isentropic compressible flow, impact pressure is"

qc = pstatic z( )1+γ2

M 2 γ +1( )2

4γ −2 γ −1( )M 2

#

$

%%%%

&

'

((((

1 γ−1( )

−1

)

*++

,++

-

.++

/++

Flight in the Vertical Plane �

Longitudinal Variables!

Longitudinal Point-Mass Equations of Motion"

V =CT cosα −CD( ) 1

2ρV 2S −mg sinγ

m≈CT −CD( ) 1

2ρV 2S −mg sinγ

m

γ =CT sinα +CL( ) 1

2ρV 2S −mgcosγ

mV≈CL12ρV 2S −mgcosγ

mVh = − z = −vz =V sinγr = x = vx =V cosγ

V = velocityγ = flight path angleh = height (altitude)r = range

•  Assume thrust is aligned with the velocity vector (small-angle approximation for α)"

•  Mass = constant"

Steady, Level Flight"

0 =CT − CD( ) 1

2ρV 2S

m

0 =CL12ρV 2S − mg

mVh = 0r = V

•  Flight path angle = 0"•  Altitude = constant"•  Airspeed = constant"•  Dynamic pressure = constant"

•  Thrust = Drag"

•  Lift = Weight"

Subsonic Lift and Drag Coefficients"

CL = CLo+ CLα

α

CD = CDo+ εCL

2

•  Lift coefficient"

•  Drag coefficient"

•  Subsonic flight, below critical Mach number "

CLo, CLα

, CDo, ε ≈ constant Subsonic!

Incompressible!

Power and Thrust"•  Propeller"

•  Turbojet"

Power = P = T ×V = CT12ρV 3S ≈ independent of airspeed

Thrust = T = CT12ρV 2S ≈ independent of airspeed

•  Throttle Effect"

T = TmaxδT = CTmaxδTqS, 0 ≤ δT ≤ 1

Typical Effects of Altitude and Velocity on Power and Thrust"

•  Propeller"

•  Turbojet"

Thrust of a Propeller-Driven Aircraft"

T = ηPηI

PengineV

= ηnet

PengineV

•  Efficiencies decrease with airspeed"•  Engine power decreases with altitude"

–  Proportional to air density, w/o supercharger"

•  With constant rpm, variable-pitch propeller"

whereηP = propeller efficiencyηI = ideal propulsive efficiencyηnetmax

≈ 0.85 − 0.9

•  Advance Ratio"

J = VnD

from McCormick!

Propeller Efficiency, ηP, and Advance Ratio, J"

Effect of propeller-blade pitch angle!

whereV = airspeed, m / sn = rotation rate, revolutions / sD = propeller diameter, m

Thrust of a Turbojet Engine"

T = mV θoθo −1#

$%

&

'(

θtθt −1#

$%

&

'( τ c −1( )+ θt

θoτ c

*

+,

-

./

1/2

−1012

32

452

62

•  Little change in thrust with airspeed below Mcrit"•  Decrease with increasing altitude"

wherem = mair + mfuel

θo =pstagpambient

"

#$

%

&'

(γ−1)/γ

; γ = ratio of specific heats ≈ 1.4

θt =turbine inlet temperature

freestream ambient temperature

"

#$

%

&'

τ c =compressor outlet temperaturecompressor inlet temperature

"

#$

%

&'

from Kerrebrock!

Performance Parameters"

•  Lift-to-Drag Ratio"

•  Load Factor"

LD = CL

CD

n = LW = L mg ,"g"s

•  Thrust-to-Weight Ratio" T W = T mg ,"g"s

•  Wing Loading" WS , N m2 or lb ft 2

Steady, Level Flight�

Trimmed CL and α"•  Trimmed lift

coefficient, CL"–  Proportional to

weight"–  Decrease with V2"–  At constant

airspeed, increases with altitude"

•  Trimmed angle of attack, α"–  Constant if dynamic pressure

and weight are constant"–  If dynamic pressure decreases,

angle of attack must increase"

W =CLtrimqS

CLtrim=1qW S( ) = 2

ρV 2 W S( ) = 2 eβh

ρ0V2

#

$%

&

'( W S( )

αtrim =2W ρV 2S −CLo

CLα

=

1qW S( )−CLo

CLα

Thrust Required for Steady, Level Flight"•  Trimmed thrust"

Ttrim = Dcruise =CDo

12ρV 2S

"

#$

%

&'+ε

2W 2

ρV 2S

•  Minimum required thrust conditions"

∂Ttrim∂V

= CDoρVS( ) − 4εW

2

ρV 3S= 0Necessary Condition

= Zero Slope!

Parasitic Drag! Induced Drag!

Necessary and Sufficient Conditions for Minimum

Required Thrust "

∂Ttrim∂V


2

ρV 3S= 0

Necessary Condition = Zero Slope!

Sufficient Condition for a Minimum = Positive Curvature when slope = 0!

∂ 2Ttrim∂V 2 = CDo

ρS( ) + 12εW2

ρV 4S> 0

(+)" (+)"

Airspeed for Minimum Thrust in Steady, Level Flight"

•  Fourth-order equation for velocity"–  Choose the positive root"

∂Ttrim∂V


2

ρV 3S= 0

VMT =2ρWS

"

#$

%

&'

εCDo

•  Satisfy necessary condition" V 4 =

4εCDo

ρ2#

$%%

&

'(( W S( )2

P-51 Mustang Minimum-Thrust Example"

VMT =2ρWS

"

#$

%

&'

εCDo

=2ρ1555.7( ) 0.947

0.0163=76.49ρ

m / s

Wing Span = 37 ft (9.83m)Wing Area = 235 ft 2 (21.83m2 )Loaded Weight = 9,200 lb (3, 465 kg)CDo

= 0.0163ε = 0.0576W / S = 39.3 lb / ft 2 (1555.7 N / m2 )

Altitude, mAir Density, kg/m^3 VMT, m/s

0 1.23 69.112,500 0.96 78.205,000 0.74 89.1510,000 0.41 118.87

Airspeed for minimum thrust!

Lift Coefficient in Minimum-Thrust

Cruising Flight"

VMT =2ρWS

"

#$

%

&'

εCDo

CLMT=

2ρVMT

2WS

"

#$

%

&'=

CDo

ε

•  Airspeed for minimum thrust"

•  Corresponding lift coefficient"

Power Required for Steady, Level Flight"

•  Trimmed power"

Ptrim =TtrimV = DcruiseV = CDo

12ρV 2S

"

#$

%

&'+2εW 2

ρV 2S)

*+

,

-.V

•  Minimum required power conditions"

∂Ptrim∂V

= CDo

32ρV 2S( ) − 2εW

2

ρV 2S= 0

Parasitic Drag! Induced Drag!

Airspeed for Minimum Power in Steady,

Level Flight"

•  Fourth-order equation for velocity"–  Choose the positive root"

VMP =2ρWS

"

#$

%

&'

ε3CDo

•  Satisfy necessary condition"

∂Ptrim∂V

= CDo

32ρV 2S( ) − 2εW

2

ρV 2S= 0

•  Corresponding lift and drag coefficients"

CLMP=

3CDo

εCDMP

= 4CDo

Achievable Airspeeds in Cruising Flight"

•  Two equilibrium airspeeds for a given thrust or power setting"–  Low speed, high CL, high α#–  High speed, low CL, low α#

•  Achievable airspeeds between minimum and maximum values with maximum thrust or power#

Back Side of the Thrust Curve"

Achievable Airspeeds for Jet in Cruising Flight"

Tavail =CDo

12ρV 2S

"

#$

%

&'+2εW 2

ρV 2S

CDo

12ρV 4S

"

#$

%

&'−TavailV

2 +2εW 2

ρS= 0

V 4 −TavailV

2

CDoρS

+4εW 2

CDoρS( )2

= 0

•  Thrust = constant#

•  Solutions for V can be put in quadratic form and solved easily#

€

x ≡V 2; V = ± x

ax 2 + bx + c = 0

x = −b2

±b2$

% & '

( ) 2

− c , a =1

•  4th-order algebraic equation for V#

•  With increasing altitude, available thrust decreases, and range of achievable airspeeds decreases"

•  Stall limitation at low speed"•  Mach number effect on lift and drag increases thrust required at high speed"

Thrust Required and Thrust Available for a Typical Bizjet"

Typical Simplified Jet Thrust Model!

Tmax (h) = Tmax (SL)ρ−nh

ρ(SL), n < 1

= Tmax (SL)ρ−βh

ρ(SL)$

%&

'

()

x

≡ Tmax (SL)σx

where

σ =ρ−βh

ρ(SL), n or x is an empirical constant

Thrust Required and Thrust Available for a Typical Bizjet"

Typical Stall!Limit!

Maximum Lift-to-Drag Ratio"

CL( )L /Dmax =CDo

ε= CLMT

LD = CL

CD=

CL

CDo+ εCL

2

∂ CLCD

( )∂CL

=1

CDo+ εCL

2 −2εCL

2

CDo+ εCL

2( )2= 0

•  Satisfy necessary condition for a maximum"

•  Lift-to-drag ratio"

•  Lift coefficient for maximum L/D and minimum thrust are the same"

Airspeed, Drag Coefficient, and Lift-to-Drag Ratio for L/Dmax"

VL /Dmax = VMT =2ρ

WS

"#$

%&'

εCDo

CD( )L /Dmax = CDo+ CDo

= 2CDo

L / D( )max =CDo

ε

2CDo

=1

2 εCDo

•  Maximum L/D depends only on induced drag factor and zero-α drag coefficient"

Airspeed!

Drag !Coefficient!

Maximum !L/D!

Lift-Drag Polar for a Typical Bizjet"

•  L/D equals slope of line drawn from the origin"–  Single maximum for a given polar"–  Two solutions for lower L/D (high and low airspeed)"–  Available L/D decreases with Mach number"

•  Intercept for L/Dmax depends only on ε and zero-lift drag"

Note different scales for lift and drag!

P-51 Mustang Maximum L/D

Example"

VL /Dmax = VMT =76.49ρ

m / s

Wing Span = 37 ft (9.83m)Wing Area = 235 ft (21.83m2 )Loaded Weight = 9,200 lb (3, 465 kg)CDo

= 0.0163ε = 0.0576W / S = 1555.7 N / m2

CL( )L /Dmax =CDo

ε= CLMT

= 0.531

CD( )L /Dmax = 2CDo= 0.0326

L / D( )max =1

2 εCDo

= 16.31

Altitude, mAir Density, kg/m^3 VMT, m/s

0 1.23 69.112,500 0.96 78.205,000 0.74 89.1510,000 0.41 118.87

Optimal Cruising Flight�

Cruising Range and Specific Fuel Consumption"

0 =CT − CD( ) 1

2ρV 2S

m

0 =CL12ρV 2S − mg

mVh = 0r = V

•  Thrust = Drag"

•  Lift = Weight"

•  Specific fuel consumption, SFC = cP or cT"

•  Propeller aircraft"•  Jet aircraft"

wf = −cPP proportional to power[ ]wf = −cTT proportional to thrust[ ]wherewf = fuel weight

€

cP :kg skW

or lb sHP

cT :kg skN

or lb slbf

Breguet Range Equation for Jet Aircraft"

drdw

=dr dtdw dt

=rw=

V−cTT( )

= −VcTD

= −LD"

#$

%

&'VcTW

dr = − LD"

#$

%

&'VcTW

dw

•  Rate of change of range with respect to weight of fuel burned"

•  Range traveled"

Range = R = dr0

R

∫ = −LD#

$%

&

'(VcT

#

$%

&

'(

Wi

Wf

∫ dww

Louis Breguet, 1880-1955!

Breguet Range Equation for Jet Aircraft"

•  For constant true airspeed, V = Vcruise!

R = − LD"

#$

%

&'VcruisecT

"

#$

%

&'ln w( )Wi

Wf

=LD"

#$

%

&'VcruisecT

"

#$

%

&'ln

Wi

Wf

"

#$$

%

&''=

CL

CD

"

#$

%

&'VcruisecT

"

#$

%

&'ln

Wi

Wf

"

#$$

%

&''

Dassault !Etendard IV!

Maximum Range of a Jet Aircraft Flying at

Constant True Airspeed"

∂R∂CL

=∂ VCL CD( )

∂CL

= 0 leading to CLMR=

CDo

3ε

•  For given initial and final weight, range is maximized when product of V and L/D is maximized"

•  Breguet range equation for constant V = Vcruise"

R = VcruiseCL

CD

!

"#

$

%&1cT

!

"#

$

%&ln

Wi

Wf

!

"##

$

%&&

! Vcruise as fast as possible"! ρ as small as possible"! h as high as possible"

CLMR=

CDo

3ε: Lift Coefficient for Maximum Range

Maximum Range of a Jet Aircraft Flying at Constant True Airspeed"•  Because weight decreases as fuel burns, and V is

assumed constant, altitude must increase to hold CL constant at its best value (�cruise-climb�)"

CLMRq t( )S =W t( )⇒

q t( ) = 12ρ t( )Vcruise2 =

W t( )S

"

#$

%

&'

3εCDo

⇒

ρ(t) = ρoe−βh(t ) =

2Vcruise2

W (t)S

$

%&'

()3εCDo

⇒

h W t( ),Vcruise!" #$

Maximum Range of a Jet Aircraft Flying at

Constant Altitude"

•  Range is maximized when "

Range = − CL

CD

"

#$

%

&'1cT

"

#$

%

&'

2CLρSWi

Wf

∫ dww1 2

=CL

CD

"

#$$

%

&''2cT

"

#$

%

&'

2ρS

Wi1 2 −Wf

1 2( )

Vcruise t( ) =2W t( )

CLρ hfixed( )S•  At constant altitude"

CL

CD

!

"##

$

%&&= maximum and ρ =minimum

h = maximum

()*

+*

!  Cruise-climb usually violates air traffic control rules"

!  Constant-altitude cruise does not"!  Compromise: Step climb from

one allowed altitude to the next"

Next Time: �Gliding, Climbing, and

Turning Flight��

Reading �Flight Dynamics, 130-141, 147-155 �

Virtual Textbook, Parts 6,7 �


Air Data Probes"Redundant pitot tubes on F-117"

Total and static temperature probe"

Total and static pressure ports on Concorde"

Stagnation/static pressure probe"

Redundant pitot tubes on Fouga Magister"

Cessna 172 pitot tube"

X-15 �Q Ball�"

Flight Testing Instrumentation"•  Air data measurement far from

disturbing effects of the aircraft"

z =

pstagnation ,Tstagnationpstatic ,Tstatic

αB

βB

#

$

%%%%%

&

'

(((((

=

Stagnation pressure and temperatureStatic pressure and temperature

Angle of attackSideslip angle

#

$

%%%%%

&

'

(((((

Trailing Tail Cones for Accurate Static Pressure Measurement"

•  Air data measurement far from disturbing effects of the aircraft"

Air Data Instruments (�Steam Gauges�)"

Altimeter"

1 knot = 1 nm / hr= 1.151 st.mi. / hr = 1.852 km / hr

Calibrated Airspeed Indicator"

Modern Aircraft Cockpit Panels"Cirrus SR-22 Panel" Boeing 777 �Glass Cockpit�"

Air Data Computation for Subsonic Aircraft"


Air Data Computation for Supersonic Aircraft"


The Mysterious Disappearance of Air France Flight 447 (Airbus A330-200)"

http://en.wikipedia.org/wiki/AF_447!BEA Interim Reports, 7/2/2009 & 11/30/2009!http://www.bea.aero/en/enquetes/flight.af.447/flight.af.447.php!

Suspected Failure of Thales Heated Pitot Probe!

�Visual examination showed that the airplane was not destroyed in flight; it appears to have struck the surface of the sea in level flight with high vertical acceleration.�!

Achievable Airspeeds in Propeller-Driven

Cruising Flight"

Pavail = TavailV

V 4 −PavailVCDo

ρS+

4εW 2

CDoρS( )2

= 0

•  Power = constant#

•  Solutions for V cannot be put in quadratic form; solution is more difficult, e.g., Ferrari�s method#

aV 4 + 0( )V 3 + 0( )V 2 + dV + e = 0

•  Best bet: roots in MATLAB#

Back Side of the Power

Curve"

Breguet Range Equation for Propeller-Driven

Aircraft"

drdw

=rw=

V−cPP( )

= −V

cPTV= −

VcPDV

= −LD"

#$

%

&'1

cPW

•  Rate of change of range with respect to weight of fuel burned"

•  Range traveled"

Range = R = dr0

R

∫ = −LD#

$%

&

'(1cP

#

$%

&

'(

Wi

Wf

∫ dww

Breguet 890 Mercure!

Breguet Range Equation for Propeller-Driven Aircraft"

•  For constant true airspeed, V = Vcruise!

R = − LD"

#$

%

&'1cP

"

#$

%

&'ln w( )Wi

Wf

=CL

CD

"

#$

%

&'1cP

"

#$

%

&'ln

Wi

Wf

"

#$$

%

&''

•  Range is maximized when "

CL

CD

!

"#

$

%&= maximum= L

D( )max

Breguet Atlantique!

P-51 Mustang Maximum Range

(Internal Tanks only)"

W =CLtrimqS

CLtrim=1qW S( ) = 2

ρV 2 W S( ) = 2 eβh

ρ0V2

#

$%

&

'( W S( )

R = CL

CD

!

"#

$

%&max

1cP

!

"#

$

%&ln

Wi

Wf

!

"##

$

%&&

= 16.31( ) 10.0017!

"#

$

%&ln

3,465+6003,465

!

"#

$

%&

=1,530 km (825 nm( )

Gliding, Climbing, and Turning Flight Performance

Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2012!



•  Flight envelope"•  Minimum glide angle/rate"•  Maximum climb angle/rate"•  V-n diagram"•  Energy climb"•  Corner velocity turn"•  Herbst maneuver"

The Flight Envelope�

Flight Envelope Determined by Available Thrust"

•  Flight ceiling defined by available climb rate"–  Absolute: 0 ft/min"–  Service: 100 ft/min"–  Performance: 200 ft/min" •  Excess thrust provides the

ability to accelerate or climb"

•  Flight Envelope: Encompasses all altitudes and airspeeds at which an aircraft can fly "–  in steady, level flight "–  at fixed weight"

Additional Factors Define the Flight Envelope"

•  Maximum Mach number"•  Maximum allowable

aerodynamic heating"•  Maximum thrust"•  Maximum dynamic

pressure"•  Performance ceiling"•  Wing stall"•  Flow-separation buffet"

–  Angle of attack"–  Local shock waves"

Piper Dakota Stall Buffet"http://www.youtube.com/watch?v=mCCjGAtbZ4g!

Boeing 787 Flight Envelope (HW #5, 2008)"

Best Cruise Region"

Gliding Flight�

D = CD12ρV 2S = −W sinγ

CL12ρV 2S =W cosγ

h = V sinγr = V cosγ

Equilibrium Gliding Flight" Gliding Flight"•  Thrust = 0"•  Flight path angle < 0 in gliding flight"•  Altitude is decreasing"•  Airspeed ~ constant"•  Air density ~ constant "

tanγ = −D

L= −

CD

CL

=hr=dhdr; γ = − tan−1 D

L#

$%

&

'(= −cot−1

LD#

$%

&

'(

•  Gliding flight path angle "

•  Corresponding airspeed "

Vglide =2W

ρS CD2 + CL

2

Maximum Steady Gliding Range"

•  Glide range is maximum when γ is least negative, i.e., most positive"

•  This occurs at (L/D)max "

Maximum Steady Gliding Range"

•  Glide range is maximum when γ is least negative, i.e., most positive"

•  This occurs at (L/D)max "

tanγ =hr= negative constant =

h−ho( )r − ro( )

Δr = Δhtanγ

=−Δh− tanγ

= maximum when LD= maximum

γmax = − tan−1 D

L#

$%

&

'(min

= −cot−1 LD#

$%

&

'(max

Sink Rate "•  Lift and drag define γ and V in gliding equilibrium"

D = CD12ρV 2S = −W sinγ

sinγ = − DW

L = CL12ρV 2S =W cosγ

V =2W cosγCLρS

h =V sinγ

= −2W cosγCLρS

DW$

%&

'

()= −

2W cosγCLρS

LW$

%&

'

()DL

$

%&

'

()

= −2W cosγCLρS

cosγ 1L D$

%&

'

()

•  Sink rate = altitude rate, dh/dt (negative)"

•  Minimum sink rate provides maximum endurance"•  Minimize sink rate by setting ∂(dh/dt)/dCL = 0 (cos γ ~1)"

Conditions for Minimum Steady Sink Rate"

h = − 2W cosγCLρS

cosγ CD

CL

$

%&

'

()

= −2W cos3γ

ρSCD

CL3/2

$

%&

'

() ≈ −

2ρWS

$

%&

'

()CD

CL3/2

$

%&

'

()

CLME=

3CDo

εand CDME

= 4CDo

L/D and VME for Minimum Sink Rate"

VME =2W

ρS CDME

2 + CLME2≈

2 W S( )ρ

ε3CDo

≈ 0.76VL Dmax

LD( )

ME=14

3εCDo

=32

LD( )

max≈ 0.86 L

D( )max

L/D for Minimum Sink Rate"•  For L/D < L/Dmax, there are two solutions"•  Which one produces minimum sink rate?"

LD( )

ME≈ 0.86 L

D( )max

VME ≈ 0.76VL Dmax

Gliding Flight of the P-51 Mustang"

Loaded Weight = 9,200 lb (3,465 kg)

L /D( )max =1

2 εCDo

=16.31

γMR = −cot−1 L

D$

%&

'

()max

= −cot−1(16.31)= −3.51°

CD( )L/Dmax = 2CDo= 0.0326

CL( )L/Dmax =CDo

ε= 0.531

VL/Dmax =76.49ρ

m / s

hL/Dmax =V sinγ = −4.68ρm / s

Rho=10km = 16.31( ) 10( ) =163.1 km

Maximum Range Glide"

Loaded Weight = 9,200 lb (3,465 kg)S = 21.83m2

CDME= 4CDo

= 4 0.0163( ) = 0.0652

CLME=

3CDo

ε=

3 0.0163( )0.0576

= 0.921

L D( )ME =14.13

hME = −2ρWS

$

%&

'

()CDME

CLME3/2

$

%&&

'

())= −

4.11ρm / s

γME = −4.05°

VME =58.12ρ

m / s

Maximum Endurance Glide"

Climbing Flight�

•  Rate of climb, dh/dt = Specific Excess Power "

Climbing Flight"

V = 0 =T −D−W sinγ( )

m

sinγ =T −D( )W

; γ = sin−1T −D( )W

γ = 0 =L −W cosγ( )

mVL =W cosγ

h = V sinγ = VT − D( )W

=Pthrust − Pdrag( )

W

Specific Excess Power (SEP) = Excess PowerUnit Weight

≡Pthrust − Pdrag( )

W

•  Flight path angle " •  Required lift"

•  Note significance of thrust-to-weight ratio and wing loading"

Steady Rate of Climb"

h =V sinγ =V TW"

#$

%

&'−

CDo+εCL

2( )qW S( )

*

+,,

-

.//

€

L = CLq S = W cosγ

CL =WS

#

$ %

&

' ( cosγ

q

V = 2 WS

#

$ %

&

' ( cosγCLρ

h =V TW!

"#

$

%&−

CDoq

W S( )−ε W S( )cos2 γ

q

*

+,

-

./

=VT h( )W

!

"#

$

%&−

CDoρ h( )V 3

2 W S( )−2ε W S( )cos2 γ

ρ h( )V

•  Climb rate "

•  Necessary condition for a maximum with respect to airspeed"

Condition for Maximum Steady Rate of Climb"

h =V TW!

"#

$

%&−

CDoρV 3

2 W S( )−2ε W S( )cos2 γ

ρV

∂ h∂V

= 0 = TW"

#$

%

&'+V

∂T /∂VW

"

#$

%

&'

(

)*

+

,-−3CDo

ρV 2

2 W S( )+2ε W S( )cos2 γ

ρV 2

Maximum Steady "Rate of Climb: "

Propeller-Driven Aircraft"

∂Pthrust∂V

= 0 = TW"

#$

%

&'+V

∂T /∂VW

"

#$

%

&'

(

)*

+

,-•  At constant power"

∂ h∂V

= 0 = −3CDo

ρV 2

2 W S( )+2ε W S( )ρV 2

•  With cos2γ ~ 1, optimality condition reduces to"

•  Airspeed for maximum rate of climb at maximum power, Pmax"

V 4 =43!

"#$

%&ε W S( )2

CDoρ2

; V = 2W S( )ρ

ε3CDo

=VME

Maximum Steady Rate of Climb: "

Jet-Driven Aircraft"•  Condition for a maximum at constant thrust and cos2γ ~ 1"

•  Airspeed for maximum rate of climb at maximum thrust, Tmax"

∂ h∂V

= 0

0 = ax2 + bx + c and V = + x

= −3CDo

ρ

2 W S( )V 4 +

TW#

$%

&

'(V 2 +

2ε W S( )ρ

= −3CDo

ρ

2 W S( )V 2( )2 + T

W#

$%

&

'( V 2( )+ 2ε W S( )

ρ

Optimal Climbing Flight�

What is the Fastest Way to Climb from One Flight Condition to Another?" •  Specific Energy "

•  = (Potential + Kinetic Energy) per Unit Weight"•  = Energy Height"

Energy Height"

•  Could trade altitude with airspeed with no change in energy height if thrust and drag were zero"

Total EnergyUnit Weight

≡ Specific Energy =mgh + mV 2 2

mg= h +

V 2

2g≡ Energy Height, Eh , ft or m

Specific Excess Power"

dEh

dt=ddt

h+V2

2g!

"#

$

%&=

dhdt+Vg!

"#

$

%&dVdt

•  Rate of change of Specific Energy "

=V sinγ + Vg"

#$

%

&'T −D−mgsinγ

m"

#$

%

&'=V

T −D( )W

=VCT −CD( ) 1

2ρ(h)V 2S

W

= Specific Excess Power (SEP)= Excess PowerUnit Weight

≡Pthrust −Pdrag( )

W

Contours of Constant Specific Excess Power"

•  Specific Excess Power is a function of altitude and airspeed"•  SEP is maximized at each altitude, h, when" d SEP(h)[ ]

dV= 0

Subsonic Energy Climb"•  Objective: Minimize time or fuel to climb to desired altitude

and airspeed"

Supersonic Energy Climb"•  Objective: Minimize time or fuel to climb to desired altitude

and airspeed"

The Maneuvering Envelope�

•  Maneuvering envelope: limits on normal load factor and allowable equivalent airspeed"–  Structural factors"–  Maximum and minimum

achievable lift coefficients"–  Maximum and minimum

airspeeds"–  Protection against

overstressing due to gusts"–  Corner Velocity: Intersection

of maximum lift coefficient and maximum load factor"

Typical Maneuvering Envelope: V-n Diagram"

•  Typical positive load factor limits"–  Transport: > 2.5"–  Utility: > 4.4"–  Aerobatic: > 6.3"–  Fighter: > 9"

•  Typical negative load factor limits"–  Transport: < –1"–  Others: < –1 to –3"

C-130 exceeds maneuvering envelope"http://www.youtube.com/watch?v=4bDNCac2N1o&feature=related!

Maneuvering Envelopes (V-n Diagrams) for Three Fighters of the Korean War Era"

Republic F-84"

North American F-86"

Lockheed F-94"

Turning Flight�

•  Vertical force equilibrium"

Level Turning Flight"

L cosµ =W

•  Load factor"

n = LW = L mg = secµ,"g"s

•  Thrust required to maintain level flight"

Treq = CDo+ εCL

2( ) 12 ρV2S = Do +

2ερV 2S

Wcosµ

#

$%&

'(

2

= Do +2ερV 2S

nW( )2

µ :Bank Angle

•  Level flight = constant altitude"•  Sideslip angle = 0"

•  Bank angle"

Maximum Bank Angle in Level Flight"

cosµ = WCLqS

=1n=W 2ε

Treq −Do( )ρV 2S

µ = cos−1 WCLqS

$

%&

'

()= cos−1

1n$

%&'

()= cos−1 W

2εTreq −Do( )ρV 2S

*

+,,

-

.//

•  Bank angle is limited by "

€

µ :Bank Angle

CLmaxor Tmax or nmax

•  Turning rate"

Turning Rate and Radius in Level Flight"

ξ =CLqS sinµ

mV=W tanµmV

=g tanµV

=L2 −W 2

mV

=W n2 −1

mV=

Treq − Do( )ρV 2S 2ε −W 2

mV

•  Turning rate is limited by "

CLmaxor Tmax or nmax

•  Turning radius "

Rturn =

Vξ=

V 2

g n2 −1

Maximum Turn Rates"

“Wind-up turns”"

•  Corner velocity"

Corner Velocity Turn"

•  Turning radius "

Rturn =V 2 cos2 γ

g nmax2 − cos2 γ

Vcorner =2nmaxWCLmas

ρS

•  For steady climbing or diving flight"sinγ = Tmax − D

W

Corner Velocity Turn"

•  Time to complete a full circle "

t2π =V cosγ

g nmax2 − cos2 γ

•  Altitude gain/loss "Δh2π = t2πV sinγ

•  Turning rate "

ξ =g nmax

2 − cos2 γ( )V cosγ

�Not a turning rate comparison�"http://www.youtube.com/watch?v=z5aUGum2EiM!

Herbst Maneuver"•  Minimum-time reversal of direction"•  Kinetic-/potential-energy exchange"•  Yaw maneuver at low airspeed"•  X-31 performing the maneuver"

Next Time:�Aircraft Equations of Motion�

�

Reading�Flight Dynamics, 155-161 �

Virtual Textbook, Parts 8,9 �

MAE331 Lecture

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Transcript of MAE331 Lecture