Maneuvering, Aeroelasticity,

and Control InteractionsRobert Stengel, Aircraft Flight Dynamics

MAE 331, 2008

• High angle of attackand angular rates

• Nonlinearaerodynamics

• Inertial coupling

• Fuel sloshing

• Aeroelasticity

• Rudder lock

Copyright 2008 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html

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

Tactical Airplane Maneuverability

• Maneuverability parameters– Stability

– Roll rate and acceleration

– Normal load factor

– Thrust/weight ratio

– Pitch rate

– Transient response

– Control forces

• Dogfights– Preferable to launch missiles at long range

– Dogfight is a backup tactic

– Preferable to have an unfair advantage

• Air-combat sequence– Detection

– Closing

– Attack

– Maneuvers, e.g.,• Scissors

• High yo-yo

– Disengagement

Maneuvering Aircraft May Be

Rotating About All Axes

!

˙ u = X /m " gsin# + rv " qw

˙ v = Y /m + gsin$ cos# " ru + pw

˙ w = Z /m + gcos$ cos# + qu " pv

!

˙ p = IzzL + IxzN " Ixz Iyy " Ixx " Izz( )p + Ixz

2 + Izz Izz " Iyy( )[ ]r{ }q( ) ÷ IxxIzz " Ixz

2( )

˙ q = M " Ixx " Izz( )pr " Ixz p2" r

2( )[ ] ÷ Iyy

˙ r = IxzL + IxxN " Ixz Iyy " Ixx " Izz( )r + Ixz

2 + Ixx Ixx " Iyy( )[ ]p{ }q( ) ÷ IxxIzz " Ixz

2( )

!

˙ x I = cos" cos#( )u + $cos% sin# + sin% sin" cos#( )v + sin% sin# + cos% sin" cos#( )w

˙ y I = cos" sin#( )u + cos% cos# + sin% sin" sin#( )v + $sin% cos# + cos% sin" sin#( )w

˙ z I = $sin"( )u + sin% cos"( )v + cos% cos"( )w

!

˙ " = p + qsin" + rcos"( ) tan#

˙ # = qcos" $ rsin"

˙ % = qsin" + rcos"( )sec#

• Rate of change of Translational Position

• Rate of change of Angular Position

• Rate of change of Translational Velocity

• Rate of change of Angular Velocity

• Stability axes offer no advantage over original body axes, as angle ofattack and sideslip angle are continually changing

Longitudinal Motions are

Coupled to Lateral-

Directional Motions

!

F =FLon

FLat"DirLon

FLon

Lat"DirFLat"Dir

#

$ %

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Lon,F

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Effects of longitudinal perturbationson longitudinal motion

Effects of longitudinal perturbations

on lateral-directional motion

Effects of lateral-directional

perturbations on longitudinal motion

Effects of lateral-directional perturbations

on lateral-directional motion

• Linearized equations have limited application tohigh-angle/high-rate maneuvers– Steady, non-zero sideslip angle (Sec. 7.1, FD)

– Steady turn (Sec. 7.1, FD)

– Steady roll rate

Effects of Nominal Sideslip

and Angle of Attack

Effects of Nominal Sideslip

and Roll Rate

Pitch-Yaw Coupling Due

To Steady Roll Rate• Assume

– Body axes

– Roll rate = constant = po, rad/s

– Consider only pitching and yawing motions

!

"u(t) =

"#E

"#A

"#R

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(

) ) )

Elevator, deg or rad

Ailerons, deg or rad

Rudder, deg or rad

!

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"x(t) ="xLon

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% % % %

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Normal velocity, m/s

Pitch rate, rad/s

Side velocity, m/s

Yaw rate, rad/s

• Control input vector

• 4th-order dynamic model

• State vector

Pitch-Yaw Coupling Due

To Steady Roll Rate

!

FLon FLDLon

FLonLD

FLD

"

# $

%

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• 4th-order stability matrix with

– Neglible vo

– Negligible coupling aerodynamic effects

Pitch-Yaw Coupling Due

To Steady Roll Rate

!

" rolling s( ) = s# Zw( ) s#Mq( ) # uoMw[ ] s#Yv( ) s# Nr( ) + uoNv[ ]{ }

+ po2

s#Mq( ) s# Nr( ) # s# Zw( ) s#Yv( )Izz # Ixx( )Iyy

Ixx # Iyy( )Izz

# uoMw

Ixx # Iyy( )Izz

# uoNv

Izz # Ixx( )Iyy

$ % &

' &

( ) &

* &

# po4Izz # Ixx( )Iyy

Ixx # Iyy( )Izz

• 4th-order stability matrix with

– Neglible vo

– Negligible coupling aerodynamic effects

• Coupling effect is proportional to po2 and po

4

• Therefore, it is independent of the sign of po2

Effect of Steady Roll Rate on

Pitching and Yawing Roots

• Faster mode increases natural frequency

• Slower mode decreases natural frequency and may become unstable

No Damping With Typical Damping

Inertial Coupling at High Roll Rate

Conditional Instability

• Problem for “fuselage-heavy” aircraft(e.g., F-104)

• See Flight Dynamics, Chapter 7

• Root locus with varying roll rate– Values of mode eigenvalues (or roots) as

functions of the steady roll rate

– Complex root pairs indicate oscillatoryresponse

– Real roots indicate non-oscillatory response

• Parameter plot: variation of real andimaginary parts of roots with roll rate

Lockheed F-104

Dutch roll: solid

Short period: dashed

Dutch roll: solid

Short period: dashed

• Effects of steady roll rate

– “p jump” or “p acceleration”

– Pitch-yaw coupling• See Flight Dynamics, Chapter 7

Aileron Anglevs. Roll Rate

Sideslip Response toElevator vs. Roll Rate

Angle of AtackResponse to Elevatorvs. Roll Rate

Inertial Coupling at High Roll Rate

- Multiple Equilibria

North American F-100

Fairchild-Republic A-10

Tumbling, Spins, and Recovery

• Strong nonlinear effects

• Aircraft-specific control strategy for recovery

Stalls and Spins

• Sidney B. Gates, RAE: "The Spinning of Aeroplanes" (with L.W. Bryant,

1926), neutral and maneuver points, stick force per g

• Continued research on stalls and spins at NASA, USAF, and in manyother countries

NASA Langley Spin Tunnel

Tails with Negative

Dihedral (Anhedral)

• Horizontal tail belowwing's wake

• May have adverseeffect on spin

characteristics

McDonnell F-4

Spin rate, !b/2V

Yaw coefficient,Cn

Yawing Moment at High Angle of Attack

• Dynamic as well as staticeffects,e.g., hysteresis

• Random asymmetric yawingmoments

– generated by slender noseat zero sideslip angle

– may exceed rudder controlpower

F-111

F-4

F-104

Fairchild-Republic A-10

Controlling Yawing Moment at High

Angle of Attack• Sucking, blowing, or movable

strakes to control nose vortices

• X-29, F/A-18 HARV

• Vortex bursting effect on tail

Grumman X-29

McDonnell-Douglas F/A-18

McDonnell-Douglas F/A-18

Deep Stall• Pitch equilibrium point at high angle

of attack

• T-tail effect: as angle of attackincreases, tip vortices are

– not only in the plane of the tail,

– but the spanwise location is closerto the centerline due to tip stall,

increasing the effect

• Other aerodynamic origins

Learjet 23

General Dynamics F-16

Supermaneuverability• Means of forcing opponent to

overshoot

• Pugachev!s Cobra maneuver,first done in Sukhoi Su-27

• Beneficial effect of thrust-vectorcontrol (X-31)

• Mongoose maneuver (X-31)

Naval Aviation• Takeoff from steam catapult at full

power, elevator up, high g (F/A-18)

• Landing at full power, no flare, needto capture tail hook, follow preciseglide slope, pitching deck,superstructure wake turbulence(EA-6B)

• T-45 takeoff and landing

Propeller-Driven Aircraft

• Takeoff without catapult, relativelylow landing speed

• Tailhook and arresting gear

• Carrier steams into wind

• Design for storage (short tail length,folding wings) affects stability andcontrol

• Chance-Vought F4U Corsair

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

Sea Planes: Flying Boats and Hydroskis

• No need for runways

• Sea state

• Corrosion

• Added weight

• Unstick

Convair XF2Y Sea Dart

Martin XP6M Seamaster

Lockheed PBY Catalina

Marttin PBM Mariner Martin PB2M Mars

Interesting, Cancelled Naval Aircraft

Chance-Vought V-173 Grumman XF5FDouglas F6D Missileer

Rockwell International XFV-12A

Grumman XF12F

General Dynamics/McDonnell-Douglas A-12Avenger II

Direct-Lift Control-Approach

Power Compensation

• F-8 Crusader– Variable-incidence wing to

provide better pilot visibility

– Flight path control at lowapproach speeds requires throttleuse, could not be made with pitchcontrol alone

– Engine response time is slow

– Flight test of direct lift control(DLC), using the ailerons as flaps

• Approach power compensationfor A-7 Corsair II and direct liftcontrol were studied at Princetonusing our Variable-ResponseResearch Aircraft

Princeton VRA

Vought A-7

Vought F-8

Direct-Lift/Drag Control

• Direct-lift control installed onS-3A Viking– Implemented with spoilers

and beeper button

– Rigged “up” during landingto allow ± lift.

• Speed brakes on T-45AGoshawk make up for slowspool-up time of jet engine

– Derivative of BritishAerospace Hawk

– Hawk's speed brake moved tosides for carrier landing

– Idle speed increased from55% to 78% to allow moreeffective modulation viaspeed brakes

Lockheed S-3A

Boeing T-45

Ground Effect

• Close proximity to the ground

– downwash is impeded

– lift is increased

• Complex effects on lift,pitching moment, and rollingmoment (see Flight Dynamics)

• Analysis models the effect bya mirror image of the aircraft

• Analogous to wind-tunnel wallcorrections

Concorde

• Coupling of non-rigid dynamic modes with rigid-body modes

• Resonant response

– Dynamically coupled modes of motion with similar frequencies

– With light damping, oscillatory amplitudes may become large

• Coupling between longitudinal and lateral-directional effects

• Nonlinear aerodynamics

• Floating control surfaces, high hinge moments, and high

aerodynamic angles

Fuel Slosh and

Aeroelasticity

!

"˙ x aircraft

"˙ x elastic

"˙ x slosh

#

$

% % %

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=

Faircraft Felastic

aircraftFslosh

aircraft

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Fuel Shift

• Problem with partially filled fuel tank

• Single wing tank from tip to tip (A4D)

• Slow, quasi-static shift of fuel c. m.

• Rudder step throws fuel to one side,producing a strong rolling moment

(A4D)

Fuel Slosh• Dynamic oscillation of fuel center of mass,

wave motion at the fuel's surface

• Pendulum and spherical-tank analogies

• Fore-aft slosh in wing-tip tanks coupledwith the short period mode (P-80)

• Fuselage tank forward of the aircraft'scenter of mass (A4D)

– Yawing motion excites oscillatory slosh thatcouples with Dutch roll mode

• Solution: Fuel-tank baffles– Slow down fuel motion

– Force resonances to higher frequencies dueto smaller cavities

– Wing internal bracing may act as baffle

Lockheed P-80 Shooting Star(A4D)

Aeroelastic Problems of the Lockheed Electra• Prop-whirl flutter, 2 fatal accidents (1959-60)

• Structural modifications made; aircraft remained in service until 1992

• Predecessor of US Navy Orion P-3, still in service

Aeroelastic Problems are Analogous to Oscillatory Slosh

Reduced Aileron Effect

Due to Aeroelasticity

• Wing torsionreduces aileroneffect withincreasing dynamicpressure

Rolling Criterion,pb/2V

pb/2Velastic / pb/2Vrigid

Republic P-47 Thunderbolt Longitudinal

Structural Modes of

Boeing 2707-300

Supersonic Transport

Boeing 2707-300

NormalizedDeflection

Centerline station

Aeroelastic Aileron Effect of

Boeing 2707-300 Supersonic

TransportBoeing 2707-300

Elastic-to-Rigid Ratio

Mach Number

B-1 Canards for Ride Control

• Elastic modes cause severe, high-g cockpit vibration during low-altitude, high-speed flight

• Active canard surfaces reduce amplitude of the oscillations

Aeroelastic Model of Aircraft Dynamics

!

"˙ x aircraft

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#

$ %

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Faircraft Felastic

aircraft

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"˙ x aircraft = Faircraft"xaircraft + Felastic

aircraft"x elastic + Gaircraft"uaircraft

0 = Faircraft

elastic"xaircraft + Felastic"x elastic + Gelastic"uaircraft

!

"x elastic = #Felastic#1Faircraftelastic

"xaircraft +Gelastic"uaircraft[ ]

• Coupled model of rigid-body and elastic dynamics

• Assume elastic modes are fast compared to rigidmodes and are stable

• Quasi-static solution for the elastic state

Quasi-Static Aeroelastic Model of

Aircraft Dynamics: Residualization

!

"˙ x aircraft = Faircraft"xaircraft #Felastic

aircraftFelastic

#1

Faircraft

elastic"xaircraft + Gelastic"uaircraft[ ]

+ Gaircraft"uaircraft

= F'aircraft "xaircraft + G'aircraft "uaircraft

!

F'aircraft = Faircraft "Felasticaircraft

Felastic"1Faircraftelastic

G'aircraft =Gaircraft "Felasticaircraft

Felastic"1Gelastic

• Substitute for "xelastic in rigid-body equation

• Modified stability and control matrices

This procedure is called residualization

Rudder Snaking• Control-free dynamics

– Nominally symmetric controlposition

– Internal friction

– Aerodynamic imbalance

• Coupling of mechanical motionwith Dutch roll mode (see FlightDynamics)

• Solutions

– Trailing-edge bevel

– Flat-sided surfaces

– Fully powered controls

Heinkel He-162

Douglas DC-2

Boeing 314

Control Surface Buzz

Avro CF-105 buzz response to command

• Aerodynamic imbalance

• At transonic speed, normal shocksmay occur on control surface

– As surface is deflected, shocksmove differentially fore and aft

– Differential hinge moment

– May lead to a limit cycle, a self-sustained nonlinear oscillation

• Solution: Splitter-plate rudder fixesshock location for smalldeflections, eliminating theproblem but also reducing the

control's effect

• Coupling with rigid-body modes

• Fully powered controls with

actuators at the surfaces eliminatethe problem

Douglas A-4

North American FJ-4

Avro CF-105 Arrow

Rudder Lock• Rudder deflected to the stops at high

sideslip angle, in turn, trims the

aircraft at the high angle

• 3 necessary ingredients

– Low directional stability at highsideslip angle, induced by stalling ofthe vertical fin

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

– Strong negative yawing moment ofrudder

• Problematical if rudder is unpoweredand requires high foot-pedal force(e.g., “rudder float” of large WWIIaircraft)

• Solutions

– Increase high-sideslip directionalstability 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

DC-3 Rudder Lock• Solutions to rudder lock

– Increase high-sideslip directionalstability by adding a dorsal fin

– Hydraulically powered rudder

• DC-3 still had rudder lock afteradding dorsal

• Super DC-3 (R4D-4+) entirelydifferent vertical tail design

Next Time:High Speed and Altitude