A Study of aircraft lateral dynamics & ground stability

Theses

Thesis/Dissertation Collections

5-1-1998

A Study of aircraft lateral dynamics & ground

stability

Howard Brott

Follow this and additional works at:

http://scholarworks.rit.edu/theses

This Thesis is brought to you for free and open access by the Thesis/Dissertation Collections at RIT Scholar Works. It has been accepted for inclusion

Recommended Citation

&

Ground Stability

by

Howard W. Brott Ir

A Thesis Submitted

In

Partial Fulfillment

of the

Requirements for the

MASTER OF SCIENCE

In

Mechanical Engineering

Approved by:

Professor

-Dr. Kevin Kochersberger

Thesis Advisor

Professor

_

Dr. Alan Nye

Professor

_

Dr. Michael Hennessey

Professor

_

Dr. Charles Haines

Department Head

DEPARTMENT OF MECHANICAL ENGINEERING

COLLEGE OF ENGINEERING

ROCHESTER INSTITUTE OF TECHNOLOGY

Permission

Granted

I, Howard W. Brott Jr., hereby grant permission to the Wallace Memorial Library of

Rochester Institute of Technology to reproduce my thesis entitled: A Study of Aircraft

Ground Stability and Lateral Dynamics, in whole or in part. Any reproduction will not be

for commercial use or profit.

May21,1998

The_stability of anaircraft onthe_{runway is dependent}on_{many factors. In} this

thesis, amathematical modelisdevelopedthatallowstheground_stability and lateral

dynamics of an aircraft tobe analyzedwhileit is intheprocess of

taking

off orlanding.

Only

two degrees-of-freedomwillbeconsidered: lateral displacementand angular

rotation. Equationsof motion forthemodel are developed using Newtonianmechanics.

Themajor components oftheaircraftthatare included inthemodel are the main

landing

gear, thevertical _tail,andthe tailwheel. Themodelis developedinto both linearand

non-linearforms.

Comparisons are madebetween atricycle gearaircraft and ataildragger.

Simulations for boththelinearand non-linear model are performed tobetterunderstand

stability. The results ofthesesimulations are usedtocomment onthe _{applicability}ofthe

Abstract iii

TableofContents iv

ListofTables vi

ListofFigures vii

ListofSymbols xi

Chapter 1 Introduction 1

Chapter 2 Literature Review 7

Chapter 3 Tire Model 1 1

3.1 Introduction 1 1

3.2 Lateral Force 12

3.3 Linear Tire Model for Main

Landing

Gear 16

3.4 Non-Linear Tire Model for Main

Landing

Gear 16

3.5 Tire Model for Tail Wheel 26

Chapter 4 Two Degree-of-Freedom AircraftModel 28

4.1 Introduction 28

4.2 Description ofModel 29

4.2.1 Assumptions 31

4.2.2 Aircraft Parameters 32

4.2.3 Free

Body

Diagram 33

4.3 Derivation ofEquations ofMotion 35

4.4 Derivation ofTire

Slip

Angles & Vertical Tail Angle 38

4.5 Linear Model 41

4.5.1 Aircraft

Sideslip

Angle 42

4.5.2 Tire

Slip

Angle 43

4.5.4 _{External Forces} andMoments 44 4.5.5 _{Linearized Equations} ofMotion 46 4.5.6 SimulationofMain

Landing

Gear

Only

47 4.5.7 SimulationofMain

Landing

Gear & Vertical Tail

Only

.. 56

4.5.8 SimulationoftheMain

Landing

Gear,

Vertical

Tail,

& Tail Wheel 69

4.6 Non-Linear Model 76

4.6.1 Non-LinearModel Equations 76

4.6.2 SimulationofMain

Landing

Gear

Only

77 4.6.3 SimulationofMain

Landing

Gear & Vertical Tail

Only

.. 82

4.6.4 SimulationofMain

Landing

Gear,

Vertical

Tail,

& Tail Wheel 87

4.7

Summary

89

Chapter 5 Conclusion 93

References 95

Appendix A MATLAB Script Files 97

A.l MAGICFIT.m 97

A.2 MAGICERROR.m 98

A.3 NLTIRE.m 99

A.4 DOF2CONT.m 100

A.5 DOF2PARA.m 101

A.6 DOF2DEPA.m 103

A.7 DOF2LSIM.m 104

A.8 DOF2LDE.m 107

A.9 DOF2NLSI.m 108

Table3.2:NonLinear TireModel Parameters 22

Table 4.1:AircraftParameters 33

Table 4.2: Linear Tire Model Parameter 44

Table 4.3: Main

Landing

Gear Simulation Configurations 57

Table 4.4: Results ofLinear Simulation ofMain Gear

Only

52

Table 4.5: Main

Landing

Gear & Vertical Tail Simulation Configurations 57

Table_{4.6: Results of Linear Simulation ofMain Gear & Vertical Tail}

Only

60

Table 4.7: Main Gear & Tail Wheel

Cornering

Stiffness for

Simulating

Main

Gear,

Vertical

Tail,

& Tail Wheel 70

Table 4.8: Results _ofLinear SimulationofMain

Gear,

VerticalTail &

Tail Wheel 71

Table 4.9: Results ofNon-Linear Simulation ofMain Gear

Only

80

Table 4.10: Results ofNon-Linear SimulationofMain Gear &

Vertical Tail

Only

83

Table 4.1 1: Results_ofNon-Linear Simulation_ofMain

Gear,

Figure 3.1: Tire

Slip

Angle 14

Figure 3.2: Tire Lateral Forceversus

Slip

Angle 14

Figure3.3: Experimental TireData 19

Figure 3.4: Tire

Cornering

Coefficient 21

Figure 3.5: Tire Lateral Friction Coefficient. 21

Figure 3.6: TireNormalizedLateral Force 23

Figure 3.7: Reconstructed Tire Lateral Force 25

Figure 3.8: Tail Wheel

Cornering

Stiffness 27

Figure 4.1: Bicycle Model fora TaildraggerAircraft 30

Figure 4.2: Free

Body

Diagram 34

Figure 4.3: Kinematic Diagram 39

Figure 4.4: Tricycle Gear Configuration 48

Figure 4.5: Linear Lateral Displacement Response for Main Gearat

CG,

Main Gear

Only

53

Figure 4.6: Linear Angular Rotation Response for Main Gearat

CG,

Main Gear

Only

53

Figure 4. 7: Linear Lateral Displacement Response for Main Gear 0.5m

inFront_of

CG,

Main Gear

Only

54

Figure4.8: Linear Angular Rotation Response for Main Gear 0.5m

in Front_of

CG,

Main Gear

Only

54

Figure 4.9: Linear LateralDisplacementResponsefor Main Gear Position

-0.5m & Forward CG

Limit,

Main Gear & Vertical Tail

Only

61

Figure 4.10: Linear Angular Rotation Response for Main Gear Position

Figure4.11: LinearLateral

Displacement

Response for Main GearPosition

-0.5m&AftCG

Limit,

Main Gear & Vertical Tail

Only

62

Figure4.12: LinearAngularRotationResponse for Main Gear Position

-0.5m&AftCG

Limit,

Main Gear & Vertical Tail

Only

62

Figure 4.13:LinearLateral

Displacement

Responsefor Main Gear Position

atCG &_{Forward CG}

Limit,

_{Main Gear & Vertical Tail}

Only

₆₃

Figure4.14:LinearAngular RotationResponse for Main Gear Position

atCG &ForwardCG

Limit,

Main Gear & Vertical Tail

Only

63

Figure 4.15: Linear LateralDisplacementResponse for Main Gear Position

at CG & Aft CG

Limit,

_{Main Gear & Vertical Tail}

Only

₆₄

Figure 4.16: Linear Angular Rotation Response for Main Gear Position

at CG & Aft CG

Limit,

Main Gear & Vertical Tail

Only

64

Figure 4. 1 7: Linear Lateral Displacement Response for Main Gear Position 0.5min Front_ofCG & Forward CG

Limit,

Main Gear &

Vertical Tail

Only

65

Figure 4.18: Linear Angular Rotation Response for Main Gear Position 0.5m

inFront_ofCG &Forward CG

Limit,

Main Gear &

Vertical Tail

Only

65

Figure 4.19: Linear Lateral Displacement Response for Main Gear Position 0.5m in Front_ofCG &Aft CG

Limit,

Main Gear &

Vertical Tail

Only

66

Figure 4.20: Linear Angular Rotation Response for Main Gear Position

0.5min Front_ofCG &Aft CG

Limit,

Main Gear &

Vertical Tail

Only

66

Figure 4.21:

Frequency

versusMain Gear

Location,

Main Gear &

Vertical Tail

Only

68

Figure 4.22:

Damping

Ratio versusMain Gear

Location,

Main Gear &

Vertical Tail

Only

68

Figure 4.24: Linear AngularRotationResponse with Main Gearat

Full-Static

Load,

Main

Gear,

Vertical Tail & Tail Wheel 72

Figure 4.25: Linear LateralDisplacementResponse with Main Gearat

Half-Static

Load,

Main

Gear,

Vertical Tail & Tail Wheel 73

Figure 4.26: Linear AngularRotationResponse with Main Gearat

Half-Static

Load,

Main

Gear,

Vertical Tail & Tail Wheel 73

Figure 4.27:

Frequency

versusTail Wheel

Cornering

Stiffness,

Main

Gear,

Vertical Tail & Tail Wheel 75

Figure 4.28:

Damping

Ratio versusTail Wheel

Cornering

Stiffness,

Main

Gear,

Vertical Tail & Tail Wheel 75

Figure 4.29: Non-Linear Lateral Displacement Response for Main Gearat

CG,

Main Gear

Only

80

Figure 4.30: Non-Linear Angular Rotation Response for Main Gearat

CG,

Main Gear

Only

57

Figure 4.31: Non-Linear Lateral Displacement Response for Main Gear

0.5m inFront_of

CG,

Main Gear

Only

81

Figure4.32: Non-Linear Angular Rotation Response for Main Gear

0.5min Front_of

CG,

Main Gear

Only

82

Figure4.33: Non-Linear Lateral Displacement Response for Main Gear

Position-0.5m, Main Gear & Vertical Tail

Only

84

Figure 4.34: Non-LinearAngularRotation Response for Main Gear

Position-0.5m, Main Gear & Vertical Tail

Only

84

Figure4.35: Non-Linear Lateral Displacement Response for Main Gear

Position

Om,

Main Gear& Vertical Tail

Only

85

Figure4.36: Non-Linear Angular RotationResponse for Main Gear

Position 0m, Main Gear & Vertical Tail

Only

85

Figure 4.37:Non-LinearLateral Displacement Response for Main Gear

Position 0.5m, Main Gear & Vertical Tail

Only

86

Figure 4.38:Non-Linear AngularRotation Response for Main Gear

Figure 4.39: Non-Linear LateralDisplacementResponse with Main Gearat Full-Static

Load,

Main

Gear,

Vertical Tail & Tail Wheel 88

Figure 4.40: Non-Linear Angular Rotation Responsewith Main Gearat

Full-Static

Load,

Main

Gear,

Vertical Tail & Tail Wheel 89

Figure 4.41: Front Tire

Slip

Angle Resultfrom Non-Linear Model 97

a Fronttirenormalized_slip angle

Fy

Normalized

tire lateral force

6 MagicFormula intermediatevariable

W

Magic Formula intermediatevariable

O Angular_velocityofaircraft-fixed coordinate system

[rad/sec]

tt Fronttire_slip angle

[deg

or_rad]

/? _{Aircraft sideslip}angle

[deg

or_rad]

S

Difference between peaksin responses

y Angleofincidenceon vertical tail

[deg

or_rad]

fly

Tire lateral frictioncoefficient

p

Density

of air

[kg/m3]

Damping

ratio

Q.z

Angular velocityof aircraft-fixed coordinate system about z-axis

[rad/sec]

Clz

Timederivativeof angular_velocity

a Distancefrommass centerto main

landing

gear

[m]

a0 Acceleration oftheorigin ofthevehicle-fixedcoordinate system

Ap

Planformarea ofverticaltail

[m2]

ay Accelerationofvehicle mass centerin y-direction

[m/sec2]

b Distance frommass centerto aerodynamiccenter of vertical tail

[m]

Bi

MagicFormulacurvefitparameter

B3

Tire_corneringcoefficientintercept

B5

Tirelateral friction coefficientintercept

Cj

Magic Formulacurve fitparameter

C3

Tire_corneringcoefficientslope

Cs

Tirelateral friction coefficientslope

Cc

Tire corneringcoefficient

C/

Fronttire_cornering stiffness

-two tires

[N/rad]

Cf.,aii

Tailwheel_cornering stiffness

[N/rad]

Cly

Coefficientofliftofthevertical tail

[1/deg]

Ca

Tire cornering stiffness

[N/rad]

Dj

Magic Formulacurve fitparameter

E]

Magic Formulacurvefitparameter

/

fraction of weight on main

landing

gear

F Externalforce

[N]

Fy

Tirelateral force

[N]

Fy_disturbance

Side force disturbance

[N]

Fyjront-tire

Fronttire lateral force

Fyjaii-wheei

Tailwheel lateral force

[N]

Fz

Tirevertical load

[N]

g Acceleration dueto_gravity

[m/sec2]

G Linearmomentum

H Angularmomentum(aboutmass_center)

i Unitvectorinx-direction of aircraft-fixed coordinate system

Izz

Totalaircraft yaw massmoment ofintertia

[kg-m2]

j

Unitvectoriny-direction of aircraft-fixed coordinate system

k Unitvectorinz-direction of aircraft-fixed coordinate system

L Wheel base

[m]

M External moment (aboutmass _center)

m Totalaircraft mass

[kg]

Mz

Momentabout z-axis of aircraftfixedcoordinate system

r Yaw velocity

[rad/sec]

Rf

Position vectorfromaircraft mass centerto main

landing

gear

[m]

Rr

Position vectorfromaircraft mass centerto tail wheel

[m]

u Forward velocity

[m/sec]

u Time derivative offorward velocity

[m/sec2]

v _{Lateral velocity}

[m/sec]

v Time derivative oflateral velocity [m/sec

]

V Magnitudeof aircraft_velocity

[m/sec]

V0

Velocity

of aircraftfixed-coordinatesystem

[m/sec]

Vtaii

Velocity

of vertical tail

[m/sec]

Vx

Componentof_{velocity in}x-direction

[m/sec]

In

1903,

theWrightbrothersbuilttheirfirst_airplane, which

they

namedtheFlyer.

Itwas abiplane

(two-wing

_{plane)with a 12-horsepower}

(9-kilowatt)

_{gasoline engine.}

The wings,whichmeasured40feet4inches (12.29meters) from

tip

to_tip, were wooden

framescovered with cotton cloth. Thepilot

lay

inthe middle ofthelowerwing. The

engine,which was mountedto thepilot's _{right, turned two}wooden propellers located

behindthewings. Theplanehadwooden runnersthat were guided

by

rails

during

take

off and usedtoskid across theground

during

a successful landing. The Brother's had

very littlecontrol ofthe airplane whilein flight. As a_result,

they

crashed_manytimes

before

they

were ableto perfecttheir

flying

technique. Throughouttheir attemptstobuild

a

flying

machine

they

considered_{many different designs.}

They

knew thatcertain

components were_necessaryto

fly,

andthedetailsofthesecomponents wereimportantto

maintainingcontrolled

flight.1

There have been manysignificantimprovements inaviation and aircraft

construction sincethedays of clothcovered wings and wooden runners. Most aircraft are

now_verywell designedand pilotfriendly.

Maintaining

control while in flight is

hardly

a

concern_anymore, exceptinextreme conditions. Pilots can even take theirhandsoff of

thewheel or stick and still maintain stableflight.

Today,

thehardestpart about

flying

is

taking

off andlanding. The interaction between thewheels ofthe

landing

gears andthe

runway producesforces thatact _{simultaneously}withthe aerodynamic liftand

drag

forces

flightan aircraftis_essentially acrossbetween agroundvehicle and a

flying

machine.

Like anytypeof_vehicle, it isdesirabletomaintain control at alltimes.

This thesisfocuses on aircraft stability.

However,

unlike most studies

involving

aircraft, thisisa_studyoftheaircraftwhileit isstill onthe_runwayand when it is just

beginning

to take-offorjust_startingto

land,

ratherthanwhen it is in theair. The

dynamics oftheaircraft are lookedat fromthepoint of view of a ground vehicle. Since

the goal of an aircraftis to

fly,

therehas been littleconcern with thedynamic behaviorof

theaircraft whileit is still ontherunway.

Nonetheless,

there are_safetyconcerns linkedto

maneuveringan aircraftwhile it is still ontheground.

Thus,

it is importantto

fully

understand aircraft ground_{stability in} an efforttomaintain control and prevent accidents

from occurring.

Directly

relatedto this issueare the

landing

gears. The location ofthe

landing

gears influence aircraft stability. _{Yet despite this, determination} of

landing

gear

location isgoverned

by

the_geometryofthe aircraft ratherthan thestability.

An aircraft on the_{runway has}three basicmodes of performance. Vertical

dynamics refers to the vertical responseoftheaircraft to_{runway disturbances} such as

slopesorbumps. Longitudinal dynamics isconcerned withthestraight-line motion ofthe

aircraft. Factors suchas_{acceleration,}

braking,

and

drag

_playa rolein this typeof

performance.

Finally,

lateraldynamics isthestudyofthe

turning

or_{sideslipping behavior}

oftheaircraft. Tire and aerodynamicforcesarethemajoritems ofinterest in lateral

dynamics. Althougheach ofthesemodes is

important,

this thesiswill _{only focus}onthe

taking

off orlanding. The goal ofthis thesis isto maximizethelateral performance ofthe

aircraft

by

_minimizingtheseunwantedeffects.

The majorinfluenceon _{lateral dynamicsisthe}configuration oftheaircraft. Is the

aircrafta_taildragger,whichhasthemain

landing

gearin frontofthe centerof_gravity and

tailwheelbehindtheCG? Or isthe aircraftatricycle gear

design,

wherethe main

landing

gearis behindthe CGand a nose wheel is present? Recentaircraftdesigns

provide increased_{reliability, comfort,} and are easierto

fly

than theirpredecessors. In

terms of_{many design enhancements,}the tricycle

landing

gearis perhapsthemost

noteworthy.

During

the

landing

_exercise,thepilot can make a number of mistakes when

manipulatingthecontrols. Atricyclegear aircraftis more

forgiving

to thepilot. That

is,

they

are easiertosteer ontheground. Whenone leams to

fly

ina_{taildragger, it is} a

challengejustto taxiit. Thepilot_{constantly has} tobe "on therudder,"

attemptingto

maketheairplane go wherehe wants ittogo. Addalittlecrosswind, andthe exercise

sometimesbecame an adventure. Ifthe noseofthe aircraft startsto deviate fromthe

desiredpath

(yaw),

the pilothastorespondwith opposite rudder. This doesnot happen

when one learnsto

fly

in an airplane equipped with atricyclegear. An airplane with a

tricycle

landing

gearhasan inherent_tendencyto"go straight"

when

being

taxied.

Additionally,

most

training

aircraft

today

have awheel-control, not unlikethatof an

automobile.

Consequently,

one_{may have}a natural

tendency

to attemptto "drive"the

airplane. Needless to say,this _{simply does}notwork,particularly

during

gusty,crosswind

landings. It is not uncommonto see a pilot attempt

during

a

landing

exercisetocorrect

for a

"gust-induced"

drag

created

by

the

down-aileron

exacerbates the yaw whileatthe same time_raisingthe

wind_{and,thereby, removing} or_reducinganyrequired crosswind correction (sideslip).

Although one_mayget_awaywiththis when

landing

atricycle gear_aircraft, with a

taildragger, a

landing

incidentoraccidentis

highly

probable. Most

likely,

a ground

loop

willoccur. Aground

loop

is adirectional_stabilityproblem where thetail ofthe aircraft

swingstothefront oftheplane.

All oftheseproblems are causedin thesamebasic way. It hasto do withthe

place wherethe side loads are reacted out againstthe ground, and wherethatiswith

respectto the center of_gravityandthe center oflateralarea oftheaircraft. Tail wheel

aircraft are not quite as

forgiving

about_{sloppy landings}as nose wheel aircraft. Partofthe

reasonforthis has todowiththe

tendency

of an irregularobjectthat_{is passing rapidly}

through theairtoalign itselfwiththe

heavy

endin front. Asa_result,thecenter of_gravity

willdo its bestto getin frontof_anyresistanceto theaircraft's motionthatisencountered.

There are also someinertial effectshere. Thesetellusthat_{if something is stopped,} it

tends to_staystopped, and _{if something ismoving,} itprefersto

keep

_{moving in} thesame

direction. Itseems apparentthen thatinorderto solvetheproblem of_stability and

ground

looping,

one mustdesignthe aircraftin such a_way astocorrectitself

during

maneuvering, as atricycle gear aircraftdoes. Ifone_really wantsto

fly

ataildragger

aircraft,then some parameters mustbe specifiedto determinea methodforthebest

landing

gear placement.

effective_wayforengineersto designand

develop

aircraft thatmeet performance goals.

Thistypeof_study will allowtheeffects ofdesignchanges tobeevaluated

immediately

to

seeif

they

meettherequired goals.

Furthermore,

computer simulation willbeeffective in

characterizingthemodel ina_controlled, repeatable environment. Inaddition, simulation

isusedtopreventtheaircraftand pilot_{from entering into dangerous} situations where

safetyis a

key

concern. _{In manyways,} this _studymirrorsthose that aredone when

evaluating aircraftflightcharacteristics.

As onemight_{expect, thecomplexity}ofthemathematicalmodel can vary. For

instance,

a model might consider_onlyone mode of_performance,be it_vertical,

longitudinal,

or

lateral,

oritcan_{be any}combination ofthese modes. Forthis _studyof

justthe lateral dynamicsof_{the aircraft,} a simple twodegree-of-freedommodel,known as

thebicyclemodel, isused. Regardless ofits simplicity, the twodegree-of-freedom model

canbe veryuseful in

demonstrating

the interaction ofthemajor parameters such as tire

properties and center of_{gravity location.}

Chapter 2 is a reviewofthe literaturethatis available on aircraft_stabilityand

vehicle

dynamics,

sincethis _{study is}a combination ofboth.

Chapter 3aims to describethebehavior ofthe tiresandtheforces

they

produce.

Also in thischapter aredescriptionsofthetwo tiremodels used inthis thesis. Both a

lineartire model and a non-lineartiremodel are usedinthis study. The non-linear model

is basedon a method calledtire datanon-dimensionalization,which was developed

by

Chapter 4 dealswith the twodegree-of-freedomaircraft model. The bicycle

model isdescribedindetail and several _simplifyingassumptions are made. The

equations ofmotion are then developedfortheentire model. Atthis point, the model is

broken intopieces to_studythe effects of each component oftheaircraftsuch as its main

landing

gear,vertical_tail,andtail wheel. Oncetheresults havebeen established and

verified, themodel isdeveloped intwoforms: linear and non-linear. Simulationis

performed foreach modelfora_varietyof aircraft component configurations.

Furthermore,

these simulationsare expandedto _{include modeling} ofthe aircraft when the

wheels arenot_{entirely in}contact with _{the runway,}as

during

take-offorlanding. The

results are then compared and conclusions are made_regardingthe _suitabilityand_quality

ofthe linearmodel. Attemptsare also made to stabilizethe aircraft given certain

Aircraft have beenstudied_extensively sincethosefirst days whentheWright

Brothers flew. Since_then,

they

have beenstudied_{in every}possible _waytoimprove their

performance and

handling

characteristicswhile inflight.

Many

mathematical simulations

have been done as well as experimental

testing

inwindtunnels and actualflight. The

interactionbetween the aircraft andthe_surrounding airinconjunction withthe control

surfaces _{is usually}the mainfocus ofthese studies. Asof_yet, there has been little

concern with theground

handling

_capability and performance of aircraft. The interaction

between the_runway andthe tires _alongwith thecontrol surfacesisnot as well known as

the interaction in flight. Performanceon theground is justnotas important as

performance inthe air. After_all, aircraft spend_onlya small percentage oftheir timeon a

runway. As such, the topicofthisthesis isa_relativelynew concept.

Becausethereisno direct literatureonthe ground

handling

_capabilityofaircraft,

threedifferenttopics were researched and broughttogether tocreate this study. The first

topic isaircraftflight stability. Some aspects ofthe aircraft's performance in flightcan

becarriedoverto ground performance. The secondtopic ofinterest is vehicle dynamics.

An aircraft onthe_runwayexhibits behaviorsthatare insome ways similarto ground

vehicles such as cars ortrucks. The finaltopicofinterest deals withtireforcesand

modelingoftirebehavior.

Oneoftheproblemsthatwasbeyondthe _graspofaeronauticalengineers inthe

early 20th

century wasthe lack of_{understanding}ofthe _{relationship between stability}and

uncommonforexperimentstobe done with uncontrolledhand-launched gliders.

Throughthese_experiments,

investigators

_discoveredthata gliderhadtobe

inherently

stableto

fly

successfully. A fewofthe_earlypioneersthatcontributedto the _stability

theory

wereAlbert ZahmfromtheUnited

States,

Alphonse Penaud from

France,

and

Frederick Lanchester fromEngland. Zahmpresented a paperin 1893 that_correctly

outlinedthe requirementsforstatic stability. He studied an airplane

descending

at

constant speed toanalyzethe conditions_{necessary for obtaining} a stable equilibrium.

Zahm's conclusion was that thecenter of_{gravity had}tobe in frontoftheaerodynamic

force.2

Otto Lilienthal of

Germany,

and Octave ChanuteandSamuel Pierpont

Langley

of

theUnited States made other significant contributionstoaircraft performance. Lilienthal

wasknown for his workon

human-carrying

gliders. He mainly dealt withtheproperties

of camberin wings. His designswere_statically stablebut had very littlecontrol

capability. As a_result, Lilienthaldied in a glider crash. Octave Chanute experimented

withbiplane and multiplane wings. Healsoincorporated a vertical tail_{for steering}and

controlsto adjustthe wingsto maintain equilibrium.

Langley,

a_secretary atthe

Smithsonian

Institution,

wasthefirstoneto experimentwithheavier-than-airpowered

flight. His initialwork consisted of_collecting and_examiningaerodynamic data.

Finally,

theWright Brothersmade

history

in 1903. Theseearlier aviators

influencedmuch oftheWright's work. The Wright's

designs,

ontheother

hand,

were

Aircraft_stability is a_{rapidly growing field}of study. The introduction of new

control_surfaces,automatic _controls, and radical _geometrymakes it difficultto get a

complete picture of_exactlywhattranspires inthe air.

However,

there are sources that

allowthereaderto get ageneral_{understanding} offlightstability. One suchbook was

written

by

a professorfrom the

University

ofNotre Dame's DepartmentofAerospace

and_{Mechanical Engineering. Flight}

Stability

_andAutomatic Control,2

by

_{Dr. Robert C.}

Nelson,

was _originallywrittenin 1989 andhassincebeenrevised. Nelson'swork covers

static _stability, aircraftequationsof_motion, dynamic _stability,

flying

or

handling

qualities, and automatic controltheory. Anothergood source foranalysisof aircraft is

Airplane Flight DynamicsandAutomatic Flight Controls

by

Jan Roskam.3

Roskam

develops themathematical modelsforthe aerodynamic andthrustforces and moments

thatact on an airplane. Healso reviews theproperties of

lifting

surfacesbefore

developing

the_steady state equations of motion of airplanes.

Finally,

the perturbed

equations of motion arediscussed.

Severalotherbooks have beenwritten thatcoverthesubject of aircraft stability.

Among

theseareAirplane

Stability

and Control: A

History

_ofthe Technologies That

Made Aviation Possible

(Abzug,

1997),4Aircraft Dynamic

Stability

andResponse

(Babister,

1986),5 Dynamics of Flight:

Stability

andControl

(Etkin,

1996),6

Flight

Stability

andControl

(Hacker,

1970),7

Performanceand

Stability

_ofAircraft

(Russell,

1996),8

andIntroduction toAircraft Flight Dynamics

(Schmidt,

1998).9

Vehicle dynamicsresearchhas been going on sincethe_{early 1900's. William}

1908. _{Another early} pioneerinthisfield wasMaurice Olley. His paper"Suspension

andHandling"covers theresearch of vehicle dynamics_upto 1937.' '

During

the_early

1960's,

Olley

summarizedhisvast knowledgeofsuspension systems and

handling

ina

series ofpapersnow known as_"Olley'sNotes".12,13

The

Millikens,

Williamand

Douglas,

didsome more recent work inthefield of

vehicledynamics. Intheir

book,

Race Car VehicleDynamics,14

they

cover_everything

fromtirebehaviorto suspensions in regardstovehicle _stabilityand control.

They

even

statein thepreface thataircraft_{engineering has been} the

key

to_{understanding} vehicle

dynamics.

The

Society

ofAutomotive Engineerspublished a number of papersin Car

Suspension Systemsand Vehicle Dynamics(SP-878)}5 thatcover suspensiondesigns and

steering systems.

They

alsodiscussanalysis techniques and effectivetest methods.

Another good referencefor both tirebehaviorand vehicledynamics is J Y.

Wong's

Theory

_ofGround Vehicles}

Wong

discussesthe mechanics of pneumatic tires

and some practical methods for predictingtire behavior. Healso deals withthe analysis

and prediction of road vehicle performance.

Otherreferences_coveringthe subject of vehicledynamics include Vehicle

Dynamics

(Ellis,

1969),17 Fundamentals_ofVehicleDynamics

(Gillespie, 1992),

18

Modeling

ofRoad Vehicle Lateral Dynamics

(Keifer, 1996),

19

Fundamentals_ofVehicle

Dynamics

(Mola,

1969),20 and_{Analysis of Tire Lateral Force} andInterpretation_of

3.1

Introduction

Tires playanimportantrolein

determining

aircraftbehaviorwhile it is onthe

ground. Thetires arethe_primarysourceoftheforces andtorquesthatprovidethecontrol

and_stabilityor

handling

oftheaircraft

(handling

_commonlyrefersto thedirectional

stabilityand_{controllability} of avehicle). The forces developed

by

the tire will affectthe

aircraftin a number of ways.

Clearly,

the tires supportthe aircraft weightinaddition to

anyother verticalforcesthat

develop

such asthoseduetoroadbanking. The interaction

betweenthetires andtheground_supplythe_tractive,

braking,

and_{cornering forces for}

maneuvering. Thetires also providetheforces usedfor controlling and_stabilizingthe

aircraft and_{for resisting}external disturbances. All ofthe_{forces acting}on an aircraft are

appliedthrough _{its tires,} withtheexception of aerodynamic and gravitational forces.

Therefore,

to _accuratelymodel thedynamicsofan aircraft onthe_{ground, it is necessary}

to havean appropriate depictionoftire behavior. Thischapterdeals withthe functionof

tires_{in producing lateral forces.}

Inthis _thesis,two tiremodels are used. Asimplelinearmodelis usedto

lay

the

foundation _{for understanding}tire

behavior,

andthena more precise non-linear model is

used for betteraccuracy. The non-linear modelis also developed because itsupports a

wide_varietyof applications. The requirements of atypical tiremodel will_vary

depending

on which aspects of vehicle performance are

being

modeled andthe _accuracy

andthree momentcomponents

(pitching,

_yawing, and_{rolling) acting} on atire dueto its

interactionwiththeground. Ifacomplete model of aircraft grounddynamics isrequired,

then all six ofthesecomponentsmustbe includedto_accuratelymodel theeffectofthe

tires on thedynamicsoftheaircraft. Thisthesis _{is only}concernedwiththelateral

dynamicsoftheaircraft.

Therefore,

_itwill _{only discuss lateral}and rotational degrees of

freedominthehorizontalplane. This will reduceto _{only forces in}thelateral direction

andmoments aboutthe vertical axis ofthe aircraft.

3.2

Lateral Force

According

to SAE

J670,

"Vehicle Dynamics Terminology",22 alateral tireforce

originates at the "center"oftirecontact with_{the road,} lies inthe horizontal plane,and is

perpendicularto the direction inwhichthe wheel is headed ifnoinclination or camber

exists.

Strictly

speaking, thecenter oftirecontactis theorigin ofthe SAE Tire Axis

System,

which isthe intersection ofthewheel center plane with the ground plane and a

point

directly

belowthecenter ofthe upright wheel. The mechanics ofthelateral force

generationis an elaborateprocess; so elaboratethat a completediscussion ofthis process

is beyond the scopeofthis thesis.

However,

_manythorough explanationsofthe

mechanics oflateralforcegeneration canbe found in

literature.14,16'18

Thelateral

force,

Fy,

generated

by

a pneumatic tiredepends on_manyvariables such as road surface

conditions, tirecarcass construction, tread

design,

rubber_{compound,size, pressure,}

surface conditionsare verticalload and_slip angle.

Thus,

thesevariables arethe main

focusofthischapterandthisthesis.

The verticalload_actingonthetires of anaircraftisafunction oftheaircraft

weight, theslopeofthe_runwayandtheliftgenerated

by

the airplane. Thetire _slip angle

is defined

by

SAE as"the anglebetweentheX' axis andthe directionoftravel ofthe

center oftirecontact".22

Simply

put,the _slip angleis theanglebetween thedirection the

wheel_{is pointing} andthedirection it is

traveling

at_anygiven instant. The tire _slipangle

isrepresented

by

the symbol a. This definition correspondsto theSAEtireaxis

29

system. As stated_earlier, theorigin ofthis system isatthe centerofthe tirecontact

patch. TheX' axis is the intersectionoftheplane ofthe wheel and theplane ofthe

ground andis positive intheforward direction. TheZ' axisisperpendicularto theplane

ofthe roadandispositiveinthe downward direction. The T axis is in theplane ofthe

road and positiveto therightinorderto formthe right-handCartesian coordinate system.

Thetire _slipangle, lateral

force,

andtireaxis system are shownin Figure

3.1,

with a

positive _slip angleandlateral force shown.

The lateral forceproduced

by

atire isa non-linearfunction of vertical load and

slipangle, as well as other variables. Figure 3.2 showsatypical lateralforce versus _slip

angle curvefora single vertical load. Asyou can see, at _{low slip}angles the curve is

approximately linear. Inthisregionthelateral forced generateddepends primarilyonthe

tire construction, tread

design,

andtirepressure. Thereis little sliding occurring between

the tireandthe ground withinthecontact patch. Lateralforce is mainly developedas a

*X'

Vd,

Y'

Figure3.1: Tire

Slip

Angle

u u u. O

a a

i

5 ca

(

D

1

2

3

4

5

6

7

8

9

10

11 12

13

14 1

The initial slopeofthis lateral forceversus_slipangle curve iscalledthe _cornering

stiffness ofthetire anddenoted

by

thesymbol _{Ca. The cornering}stiffness is often usedas

alinearapproximationforthe _relationshipbetween lateral force and _slipangle, as you

will seelater inthischapter. Ifthe_corneringstiffness isnormalized

by dividing by

the

vertical

load,

a_{quantity known} asthe_{corneringcoefficient,}

Cc,

is found. Asexpected,

both the_corneringstiffness andthe_cornering coefficient are proportional to thevertical

load. In general, the_corneringstiffness increaseswiththevertical

load,

whilethe

corneringcoefficientdecreases. Theseparameters will beneeded as the tiremodel is

further developed.

As the _slip angle

increases,

theslopeofthelateralforce curve decreases until the

lateral forcereaches a maximum. Themaximum lateralforce is usedto findthe tire

lateral frictioncoefficient,fiy.

Dividing

the maximum value

by

thevertical load does this.

Beyondthis maximum_value, the lateral force itself begins todecrease. Inthis section of

thecurve a largerportionofthe contact patch is slidingthan atlow slip angles. Herethe

lateral force produceddepends

largely

upon theroadsurface, tire rubbercompound, and

the interface betweenthe two.

The lateral forceversus _slipangle curve shown in Figure 3.2represents

steady-state tirelateral forcecharacteristics. Dueto the_elasticity and

damping

within a

pneumatic tireit_{is actually} adynamic system withinitself. When a changein slip angle

occurs, thechange in lateral force lags behind. Although theeffects oftiredynamics can

tire, theeffects are_generallysmallforlow inputfrequencies. Tire dynamics are

neglectedinthismodel as

they

are

typically

used when_simulating extreme situations.

3.3

Linear Tire Model for Main

Landing

Gear

Asstatedin theprevious _section,theinitial slope ofthe lateral forceversus_slip

angle curvefor asingle verticalloadisthe_corneringstiffness, Ca. Thisconstant canbe

used as areasonable representation oftirebehavior under certain conditions. Inspection

ofFigure 3.2 reveals thatat small _slip anglesthe _{lateral force is nearly linear.}

Thus,

at

sufficientlysmall _slipanglesthelateral forceproduced

by

atirecan beapproximated

by

Fy=Ca

(3.1)

When thislinearapproximation is combinedwith other assumptions_regarding the

aircraft,thelateral force versus_slip angle _relationshipallowsthe aircraftto bemodeledas

alinearsystem. This _{is very beneficial}to theuserbecausesome _verywelldeveloped

analysis techniquesexist thatallow usto _study alinearmodel andlearna great deal about

aircraftlateral dynamics. Therange of_{applicability}ofthelineartiremodel isexamined

by

comparisonto thenon-linear modelin thenext chapter.

3.4

Non-Linear

Tire Model

for

Main

Landing

Gear

Asyou saw in Figure

3.2,

thelateral force versus_slipangle curve isnolonger

linearat _{high slip}angles,andthus will not _accuratelypredictthe tirelateral force. Ifwe

aforcethatis greaterthan the actualtire force.

Therefore,

it is necessary to

develop

a

non-lineartiremodel thatwill_{correctlydetermine} the tirelateral forceat high slipangles.

Severalapproachesto _modelingtirebehaviorexist. Somemodelsare_purely

empirical,basedupon curve

fitting

of_{experimentally}measuredtire data. Othermodels

are_{primarily theoretical,}with some parametersdeterminedexperimentally, such as the

stiffnessofthetire. Eachtypeof modelhas its advantages anddisadvantages. Since

experimentally measured aircrafttiredataexistsfora_varietyoftires, we will usethefirst

method to

develop

themodel.

Thetire model chosenforthis _{study is}calledtiredatanon-dimensionalization,

and wasoriginated

by

HugoRadt.14,23,24 This technique isable topredicttire_aligning

moment, longitudinal

force,

andlateral force forcombinedlateral _{slip, camber,}and

longitudinalslip.

However,

_onlythelateral force due to_{lateral slip}willbe usedin this

study. Theeffects of camber on lateral forceare

being

neglected.

Also,

ifwe assumethe

aircrafthas a constant_{forward velocity}thenit isnot_necessaryto considerlongitudinal

force. Although

tire-aligning

moments are_present,

they

are consideredto havea

negligible effect onthe overalldynamicsofthe aircraft.

Thetechnique you are aboutto see consistsoftwomain steps. The _{first step deals}

with_{preprocessing}theexperimentaltiredatato determine theparametersforthe tire

model. Thesecond_step usestheseresultstocalculatethe tirelateral force for a given

vertical loadand_slipangle. Ina vehicle dynamicssimulation,the _{first step is}

typically

done before runningthesimulation. Thesecond _{step is}thendoneat each time_step

The tiredatausedin

developing

thismodel was obtainedfrom theGoodyear Tire

& Rubber

Company

forastandard aircraft_tire, model6.00-66PRat42psi. Thetiredata

isgivenin Table 3.1 andshown inFigure 3.3. Lateral force versus _slip angle curves are

availableforverticalloadsof 1779

N,

2669

N,

3559

N,

4448

N,

and _{5338 N. The slip}

angleforeach ofthese loadsvariesfrom0

to 20.

Typically,

the lateral forceat0_slip

angleis notzeroas mightbeexpected. This is due to_conicityand/or_plysteer inthe tire.

Conicity

results fromasymmetries intireconstruction,while_ply steer resultsfromerrors

intheangles ofthebeltcordsin the tire. Bothoftheseeffects are randomin nature and

vary fromtire to tire. Theseeffects are not importantto this model, andthemanufacturer

hasadjustedthedataaccordingly.

Preprocessing

theexperimental data is done

by

_normalizingthe dataandthen

curve

fitting

thenormalizeddata. The first step in normalizingthedata isto determine

the tire_corneringcoefficient,

Cc,

at eachload. As previously _stated,the_cornering

stiffnessis theinitial slope ofthelateral force versus _slipangle curve. _{The cornering}

coefficientis thenequal to the_cornering stiffnessdivided

by

thevertical

load,

which can

be approximatedat eachload

by dividing

thelateralforceat 1 _slip angle

by

the

correspondingverticalload.

Thus,

the_corneringcoefficient at_{any load is}

Table 3.1. ExperimeritalTire Data

Slip

Angle Lateral Force @ Vertical Load

(N)

(deg)

1779 2669 3559 4448 5338

0 0 0 0 0 0

1 302 414 454 485 489

2 534 734 823 899 912

3 707 988 1125 1246 1281

4 841 1179 1370 1535 1601

5 939 1334 1575 1784 1882

6 1014 1450 1739 1993 2122

7 1072 1544 1877 2166 2335

8 1112 1615 1988 2313 2518

9 1143 1673 2077 2438 2678

10 1170 1717 2153 2544 2816

11 1188 1748 2215 2633 2936

12 1201 1775 2264 2709 3038

13 1214 1797 2304 2771 3132

14 1219 1810 2340 2825 3212

15 1228 1824 2366 2869 3278

16 1232 1833 2389 2909 3341

17 1237 1842 2406 2940 3390

18 1237 1846 2424 2967 3434

19 1241 1850 2433 2989 3474

20 1241 1855 2447 3011 3510

4000

3500

3000

1779N

2669N

3559N

4448N

5338 N

8 10 12

Slip

Angle(deg)

14 16 18 20

Figure 3.4 showsthetire_corneringcoefficient foreach load. Asyou can _{see, the}

relationship between

cornering

coefficientand vertical loadis approximately linear. For

this_tire, the_corneringcoefficient as afunctionof vertical loadcan berepresented as

CC

=

B3+C3FZ

(3.3)

Values oftheconstants

B3

and

C3

are given _{in Table 3.2. The resulting}expression can be

used topredictthe_corneringcoefficientforan _{arbitrary load}

during

a simulation.

Thenext _stepis tofindthe lateral frictioncoefficient, juy, at eachload.

Dividing

themaximumlateral force fora given vertical load

by

the vertical load itself doesthis.

Thus,

fora single verticalloadthelateral frictioncoefficient is

My=^-(3.4)

r.

The lateral frictioncoefficients foreach load are plottedin Figure 3.5. As with the

corneringcoefficients, the _{relationship between} thelateral frictioncoefficient andthe

vertical load is approximately linear forthis tire. Thelateral friction coefficient can then

beexpressedas

jUv=B5 +

C5Fz

(3.5)

Valuesoftheconstants

B5

and

C5

are givenin Table 3.2. Thisexpression can also be

0.18

0.16

SP

0.14

o

0.12

-S -1 +

g

0.08 o

^

0.06 c

E 0.04

+

o

0.02

0

1000 2000 3000 4000 5000 6000

Vertical Load_(N)

Figure 3.4: Tire

Cornering

Coefficient

0.72

0.7

o

U 0.68

-c o

o

0.66

0.64

?

^^>>

V^*

y= -1E-05x+ R2

=0.9C 0.7221

)54

>. ?

?

1000 2000 3000 4000

Verticalload

(N)

5000 6000

[image:37.573.63.510.73.215.2]

Table 3.2: Non-LinearTire Model Parameters

Parameter

_{Symbol Value}

Tire_corneringcoefficient intercept

B3

0.2117

Tire_corneringcoefficient slope

C3

Tire lateralfrictioncoefficientintercept

B5

0.7221

Tirelateral frictioncoefficient slope

c5

Magic Formulacurvefitparameter

Bi

0.7947

MagicFormulacurvefitparameter

c,

0.1901

Magic Formulacurvefitparameter

Dj

6.1197

MagicFormulacurvefitparameter

E,

1.0637

Now that the_cornering coefficientandlateral frictioncoefficient are knownat

each

load,

thenormalized _slipangle, a, canbecalculated at each datapointfrom the formula

_

Ctania)

a =

Similarly,

thenormalizedlateral

force,

Fy

, at eachdatapoint is

(3.6)

MVFZ

(3.7)

Whenthenormalized lateral force is plottedagainstthe normalized_slip angleforeach

vertical

load,

theresults lieon a single curve as shownin Figure 3.6. Thisnormalized

data is then curvefit. Thereare a number of methods usedtofitthisdata. This model

uses a popularfunction knownasthe"magic formula". Themagic

formula,

as you will

see, isa combination oftrigonometricfunctionsandhasthe _capabilityto_{accurately fit}

u

o

03

<U N

1_ O

1.2

1

0.8

0.6

0.4

0.2

0

?

|

? 1779 N;

OfifiQ M

A 3559 N

x 4448 Nl

X 5338 N

2 3 4

Normalized

Slip

Angle

Figure 3.6: Tire Normalized Lateral Force

Thenormalizedlateral force is fitto thefunction

Fy

=D,sin(e)

where

6 _{=C]atan(Bly/)}

and

W

=

(l-E,)a

₊

E^atan{Bxa)

B

(3.8)

(3.9)

(3.10)

Theparameters

Bj, C],Dj,

and

E}

mustbe determinedtoprovidethebest fit to the

normalized experimentaldata. Thecurve

fitting

is implemented inthe MATLABscript

fileand uses theMATLAB OptimizationToolboxfunction

LEASTSQ

todo a non-linear

leastsquares fit. The

LEASTSQ

_function callsthefunction MAGICERROR.mthat

computes theerrorsbetween eachdatapoint andthe curvefit function. Theparameters

B\,

C\,D\,

and

E\

arefoundtominimize the sum ofthesquares ofthese errors.The files

MAGICFIT.m andMAGICERROR.m weredeveloped

by

Joseph Kieferforusein

Modeling

of Road Vehicle Lateral

Dynamics.19

They

arelisted in Appendix A.l and

Appendix

A.2,

respectively. Values forthe curvefitparameters are givenin Table

3.2,

andthefunction isplottedin Figure 3.6 along withthenormalizeddata. As you can

clearlyseefromtheplot,an excellentfitto thedata had been obtained.

Nowthatafunction forthenormalized lateralforce interms of normalized_slip

angle is available,the tirelateral force canbecalculated _{for any}combination of vertical

loadand _slipangle.

First,

the_cornering coefficient and lateral frictioncoefficient are

calculatedfromthevertical load using Eq.

(3.3)

and (3.5).

Second,

the normalized_slip

angleis calculatedfromthe_slipangle, the_corneringcoefficient, andthelateral friction

coefficient_{using Eq. (3.6).}

Next,

the normalizedlateral forceiscalculatedfromthe

normalized_slipangle_{using Eq.}

(3.10),

(3.9),

and

(3.8)

inthisorder.

Finally,

the tire

lateral forcecanthenbe found from thenormalizedlateral

force,

thelateral friction

coefficient, andthe vertical loadas

F>=FyfiyFz

(3-11)

outputsthelateral force. Plotsoflateral force versus_slipangle from thisfunction for

verticalloads of 1779

N, 2669N,

₃₅₅₉

N,

₄₄₄₈

N,

and5338 Nare shown in Figure 3.7 as

solid lines_along withthe experimental datapoints.

Thenon-lineartire modelimplementedinthis section _accuratelyreproduces the

experimentally determined lateral forceversus_slip angle_relationshipofthe tireused in

this study. Thismodel iscapableof_predictingthelateral forceproduced

by

the tire at

high _slip angles.

Thus,

the tiremodelis suitable for inclusion ina model of aircraft

lateraldynamicswhere hightire_slip angles are obtained. Whilethis tiremodel_only

determined lateral force due to_slipangle, itcanbeextended topredict _aligningmoment

dueto _slip angle, lateral forceand _aligningmomentdueto_camber, andlongitudinal force

due tolongitudinal slip.

400 -i

350

-300

-250

-

g200-150

-a

2 100

-J

500-0 i

(

;

L

0

I 1 I I

? _{1779 N}

2669 N

A _{3559 N}

x _{4448 N} * _{5338 N}

Reconstructed i>* ^

ly* _^t^*"\

^^Jr^^T^

^X^

) 2 4 6 8 10 12 1

Slip

Angle

4 16 18 2

3.5

Tire Model for Tail Wheel

Thetailwheelforthis _studywas chosen tobe an8-V2 inch outerdiameter

by

3-inchwidth tire. Theinflationpressureis 40psi. Thismodel was also createdfrom data

provided

by

GoodyearTire & Rubber Company.

Using

thedimensions ofthe tire,

they

wereableto calculatethe_corneringstiffness fora specifiedload. The datapoints were

thenplotted and acurvefitwasapplied_using theMICROSOFT EXCEL function

TRENDLINE. Thisfunctioncalculatesaleastsquaresfitthroughthe datapoints. A

fourthorder polynomial was fitto thedatapoints as shownin Figure 3.8. Theequation

EXCELusedtofitthedatapointsis displayed in thefigure. Alsoshown istheR-Squared

value. The R-Squared value canbe interpretedas theproportion ofthe variancein y

attributableto the varianceinx. This function returns thePearson product moment

correlation_{coefficient,r,} whichis adimensionless indexthat ranges from -1.0 to 1.0

inclusiveand reflects the extent of alinear relationship between twodatasets. As you

can_see, theR-Squaredvalueforthiscurvefit is exactly 1.

Therefore,

the equation shown

represents thebestpossiblefitto the data. Thetailwheel model isusedin Chapter 4

when _calculatingtail wheel _cornering stiffnessbasedonthefractionof weight on the tail

160

O) 140

CD o

7? 120

100 Cj) r 3= ₈₀

CO

c 60

i_

CD

r 40

L.

o

O ₂₀

0

I

-?

1

y=

2E-07x3

-0.0006X2

+0.551 9x+3.8856

R2=1

\

0 200 400 600 800 1000 1200

Vertical Load

(N)

1400 1600

In_{this chapter,} we willbe

looking

at a simplified mathematical model of an aircraft.

By

_using acomputer generatedmodel we willbeableto _varyall aspects ofthe aircraft. Thisis extremelyuseful forthis _{study in}which we needtoexplorethe

difference betweenatricyclegear aircraftandataildraggeraircraft. This model will

ultimately allow usto

develop

thebasic conceptsthatgovern theresponse ofthe aircraft and controlthe stability. Fromthese concepts we willbe abletopredictthemotion ofthe aircraftasitmoves_alongthe_runway regardless ofits configuration, as well as examine the _stabilityofthe aircraft asit is in theprocess of

taking

off orlanding.

This simplified representation ofthe _aircraft,referredto as the "bicycle"model, is atwo degree-of-freedommodelthat isoften usedtoexaminethe lateralresponseof road vehicles. The model isaltered _slightlytoreflectthe_geometryoftheaircraft used inthis study. The model is developed in a generalformforataildraggeraircraft. Laterin this

chapter, youwill see that_onlyminorchangesto themodel are needed to makeit intoa

tricycle gear aircraft. As you will_see, thismodel _greatlysimplifies an aircraft.

However,

through itsusewe caninvestigatethe effects of majordesignand operational parameters on the_yawingand_sideslippingmotionsthat determinethepathand attitude ofthe aircraft. Such parametersinclude tire properties,inertia_properties, center of_gravity

Thischapter aims todescribethe twodegree-of-freedommodel in detail. To

accomplish _this, each vitalcomponentoftheaircraft will be studied untilthe modelis

complete.

First,

we willstartwithjustthemain

landing

geartoseehow its location

affects stability. Aftertheseresults have beenestablished, theverticaltail will be added

ontothemodeland thechangein theresponse will beexamined.

Finally,

the _studywill

conclude withtheaddition ofthe tailwheel to the model. Atthis point, we should

understand enough abouttheeffects of each component tobeable tomake reasonable

suggestionstoimprovethe_stabilityof ataildraggeraircraft.

4.2

Description

of

Model

Thetwodegree-of-freedommodel forthe taildraggeraircraft usedin this_{study is}

shownin Figure 4. 1. As itsname suggests,themodel is onlyconcerned withtwo typesof

variable motion: lateral velocity, v, and_yawingvelocity, r. Theforward velocity, u, is

assumedtobe constant and shall bechosen

by

theuser. Thesingle tireshown represents

thepairoftiresforwardofthecenter of_{gravity for}an aircraft with ataildragger

configuration. Thetail wheel shownisallowedto swivel.

However,

itcanbe locked into

placeto create some stiffnessthatwill resultin alateral tireforce. Thevertical tailshown

willhavea significant contributionto themotion oftheaircraft. The longitudinal

alignment ofthecomponents ofthe aircraft show_whythis iscalleda

"bicycle"

T

"--^v

^

Figure4.1: Bicycle Model fora Taildragger Aircraft

On atypical vehicle,the distance fromthecenter ofthefronttiresto thecenter of

therear tiresis termed the wheelbase,L. This is also thecase with an aircraft. Forthis

particular model ofaircraft,the tail wheel is locatedalmost

directly

below the

aerodynamic center ofthe verticaltail.

Assuming

that

they

areboth in thesamelocation

mass momentof

inertia,

Iu,

withits centerof_{gravity located}adistance afromthecenter

ofthefronttireanda_{distance b fromthetail} wheelandthe aerodynamiccenter ofthe

vertical tail.

Noticethatthefronttires on anaircraftare not capable of_rotatingabout thez-axis

as on acarortruck.

Therefore,

they

cannotbeconsideredas control inputs.

Again,

this

greatly simplifies the_model, as we need notconcern ourselveswith steer angles.

The coordinate systemusedto describethemotion ofthe aircraftisthe standard

SAE vehicle-fixed x-y-z coordinatesystem,whichtranslates and rotates withthe

aircraft.

~

Thex-axis ispositiveintheforward

direction,

they-axis ispositiveto the

right, andthez-axisis positivedown into thepage as specified

by

the right-hand rule.

The origin ofthecoordinate system isatthecenter of_gravity

(CG)

ofthe aircraft.

Motion is onlypermittedinthe_x-yplane.

4.2.1

Assumptions

Therepresentation oftheaircraft as a

"bicycle"

modeltakesinto account several

simplifyingassumptions in additionto those _previouslymentioned.

They

are:

Constant aircraft parameters

(m,

Ia

a,

b, L)

Constantaircraftforwardvelocity, u

Motion in_x-yplane_{only (ignore} vertical,rolling and_{pitchingmotions)}

Aircraftisrigid

Runway

surfaceis smooth

Ignore lateral _gravityeffects _{(ignore runway}slopein lateral

direction)

Ignore_{longitudinal gravityeffects (ignore}

runwayslopein longitudinal

direction)

Ignoreall

longitudinal

_{forces (tire}

driving/braking

forces,

tire_{rollingresistance,}

aerodynamic

drag)

Ignorefront

landing

gearsuspension systemkinematics anddynamics

Ignoretire_slipangles _resultingfrom lateraltirescrub

Ignore lateral andlongitudinalloadtransfer(verticaltireforces remain_constant)

Tirepropertiesare independent oftimeand forward velocity

Ignore tirelateral forces due tocamber, conicity,and_plysteer

Ignore tire_aligning moments

Ignoreeffects of_{longitudinal slip}ontire lateral force

Ignore tiredynamics (no

delay

in lateral force_generation)

Ignore tiredeflections

4.2.2 Aircraft Parameters

The nominal values ofthe aircraft parameters usedforthe two degree-of-freedom

modelinthis_study are given in Table 4. 1. Allparameters were obtainedthrough direct

measurements on_{the aircraft,} exceptforthemass andthe CG

location,

which canbe

is 13.6-16.8 inches fromthedatum. As a point of referenceforthe reader, the forward

mostCGposition of 13.6inches fromthe datumwaschosento measure thecenterof

gravityparameters, a and

b,

givenin Table 4.1. These distances canthenbeusedto

calculatethefractionof weighton thefrontaxle,/. The SIsystem of units isusedforall

calculations. Theunit oflengthisthemeter

(m),

andtheunitoftimeis the second(s).

Force ismeasured inthederivedunitNewton (N). Theseparameters are representative of

a production aircraft.

Table4.1: Vehicle Parameters

Parameter Symbol Value

AircraftMass m 748

kg

Yaw MomentofInertia I- 1760kg-m2

Wheelbase L 4.52m

Distance from CGto Front

Landing

Gear a 0.599 m

Distance from CGto Aerodynamic CenterofVertical Tail

& Tail Wheel b 3.92 m

Front

Landing

Gear Weight Fraction

f

0.867

CG Limits CG 13.6-16.8 in

Vertical Tail Area

K

0.693m2

CoefficientofLifton Vertical Tail

CLv

0.100/deg

4.2.3

Free

Body

Diagram

Due to the _simplifyingassumptionsin Section

4.2.1,

therearefourtypes of

external _{forces acting}ontheaircraftin the_x-yplanethat are consideredin thismodel:

fronttire lateral

force,

tail wheel lateral

force,

verticaltail

force,

and adisturbance input.

Thefronttirelateral

force,

Fyjr0nt-tire,

occurs dueto tire_slipangles asdoesthe tailwheel

lateral

force,

Fy_taii.wfieei- The vertical tail

force, Fy_,au,

isafunctionoftherelative air

Fy_disturbance,

is aninputthatacts atthe aerodynamic center oftheverticaltail, as doesthe

tailforce. Themostcommondisturbance input isa crosswindencounteredon the

runway. Theseforcesare shown_actingonthevehiclein Figure 4.2 and areconsidered

positive when _{acting in}thepositivey-direction.

T

"^disturbance

t

fc^ia

"y_front-m*av

y_tail

Figure4.2: Free

Body

Diagram

4.3

Derivation

of

Equations

of

Motion

Theequationsofmotion will be developed forthe"bicycle" model. To dothis we

use basicprinciples ofNewtonianmechanics forrigid

body

motion relativeto

translating

and_rotatingcoordinatesystems.26

This is alinearmodel with twodegrees-of-freedom

which enablesthecalculation ofthemotion variables as afunctionofthe forcesand

moments _actingontheaircraft.

Referring

toFigure 4. 1, themotion variables ofinterest

areforward_{velocity, u,} lateral velocity,v, and yaw_rate, r. Yaw _{rate, r,} isthe angular

velocityoftheaircraftarounda vertical axis_passingthrough theCG. Thevector sum of

u and vis thepath _velocity, V. Thex-axisofthe aircraft is at a

body

_{slip angle,}

/?,

with V

Forstraight-ahead motion/J=0. The_{total side}force is Y_andthe

yawingmoment about

theCG is N.

The reference axis system(aircraftaxis _system) is fixedon theaircraft with the

origin attheaircraftCG (referto section4.2). The fundamental features ofthissystem

are thatchanges inthedependentvariables are measured relativeto inertial space andthe

moments and products ofinertiaare constant. Iftheactual path ofthevehicle relativeto

theground is_required,an additional earth-fixed referencesystem isrequired.

Therigid

body

equations are obtainedfrom Newton's Second Law. Thisstates

thatthesummationof all external forces actingon a

body

is equalto the time rate of

change ofthe momentumofthe body.

Furthermore,

the summation oftheexternal

moments _actingonthe

body

is equalto the timerate of change ofthemoment of

momentumare referredtoan absolute orinertialreferenceframe. Newton's Second Law

can beexpressedinthe

following

vectorequations:

Yf=-g

dt

(4.1)

Ym=-h

^

dt

whereF andMaretheexternalforcesand moments (in vector

form)

_actingaboutthe

centerof_gravity, andGandHarethe linearand angular momenta ofthe

body

(also in

vectorform). Theassumptions ofthemodel allow us toreduceEq.

(4.1)

since_only

motion inthe_x-yplane isconsidered and all longitudinal

(x-direction)

forces are

neglected.

Thus,

we are left with

SFv

=may

^

.

(4-2)

Since thex-y-z coordinate systemis fixedto the aircraft with its origin atthe

aircraft center ofgravity, the translational _velocityoftheaircraft mass center and

rotational _velocityoftheaircraftare identicalto thoseofthex-y-zsystem. From Figure

4. 1, the velocity,

V,

oftheorigin ofthex-y-zsystem is

V=m+_vj

(4.3)

andthe angular_velocity,

Q,

is

Q.= _rk

Sincethex-y-zsystemis also_{rotating, the}unit vectors are_changingwithtime.

Thus,

the

acceleration,a, oftheorigin expressedinan inertialreference framecoincident with the

x-y-zsystemis

dV a =

+

QxV

(4.5)

dt

= (u-_vr)i+

(v

₊

wr)j

Likewise,

the angular accelerationofthex-y-z system relativeto aninertial reference

frameis

n= da

dt xyz

+Q.XQ.

(4.6)

Thus,

because we are_onlyconcerned with lateral_{motion, the}acceleration values of

interest are

a =v+ur

(4.7)

n.

=r

Thesevalues _applytoboth theaircraft-fixedx-y-z system andtothe aircraft center of

gravity.

Theexternal forces actingonthevehicle are shown in Figure 4.2. The sum ofthe

forcesin they-direction andthesum ofthemoments aboutthez-axis canthen bewritten

as

/ , "

v * _y_front-lire y_disturbance y

_tail _y_tail-wheel

(4.8)

V A/f n*F -h*F +h*(F +F )

j1Y1

,

" '

y_front-tire y_disturbance ^ y

Substitution ofEq.

(4.7)

andEq.

(4.8)

into

Eq

(4.2)

yieldsthe equationsof motion

forthe two degree-of-freedomtaildraggeraircraft:

y_front-lire + y_disturbance ~^y_tail ~ *'

y_tail-wheei

=_m\V+Ur

)

a* _F h* _F A-h*_{( F} 4- _{F \ J} r

y_front-tire u L _y_disturbance ~

u \r

y_ta.il ^*

y_tail-wheel'

~ _l zz'

Hereuis thevehicle_{forward velocity}and isconstant. The state variables arethe lateral

velocity,v, andtheyaw _velocity, r.

Fyjront.tire

isthefronttirelateral

force, Fy__isturbance

is

thedisturbance

input,

Fy_taa

isthe vertical tail

force,

and

Fy_tau.wheei

isthe tail wheel lateral

force.

Asstated,theaircraft system derived in Eq.

(4.9)

is atwo degree-of-freedom

system. This is theminimum number ofindependentcoordinates neededto describe the

motion ofthe system.

Furthermore,

Eq.

(4.9)

is a constraint equation. A systeminwhich

theconstraints are functionsofthecoordinates or coordinates and timeiscalled

holonomic. In manycasestheconstraint equations relatethe velocities orthevelocities

and coordinates. Ifthese equations canbe integratedto yield relations

involving

coordinates andtime only, the system isstill

holonomic.27

However,

the system_may not

be holonomic intheevent oflarge displacements

during

integration. Sinceeach

integration step inthis _{study is} small,the system remainsholonomic.

4.4

Derivation

of

Tire

Slip

Angles &

Vertical Tail

Angle

From Chapter

3,

thelateral forceproduced

by

atiredependsuponthevertical load

velocityand angular velocity. For_{this reason,}we must

develop

an expression fort