Lateral Directional Approximations to Aircraft · 2013-02-19 · Lateral Directional Approximations to Aircraft Lateral Directional Approximations to Aircraft Joel George Department

Lateral Directional Approximations to Aircraft

Lateral Directional Approximations

to Aircraft

Joel George

Department of Aerospace Engineering,Indian Institute of Science Bangalore,

India - 560012.

July 14, 2005

Joel George Lateral Directional Approximations to Aircraft


Introduction

Modelling the Aircraft

Point mass model → Performance AnalysisEnergy Approach

gives solutions like dive–zoom path

provides only limited information6 DoF rigid aircraft model

aircraft is flexiblestill a good representation



Introduction

Aircraft Equations of Motion

6 Dof Equations of Motion

12 nonlinear 1st order ODE6 Dynamic Equations

6 Kinematic Equations

Linearize the equations about an equillibrium point



Introduction




3 Force Equations3 Moment Equations

6 Kinematic Equations




Introduction




3 Force Equations3 Moment Equations

6 Kinematic Equations3 Euler Angle Equations3 Navigation Equations




Introduction


Decoupling of Dynamics

Linearized aircraft dynamics can be decoupled asLongitudinal Dynamics

motions in the plane of symmetry

Lateral-Directional Dynamicsmotions out of plane of symmetry



Introduction




Phugoid modeShort Period mode

Lateral-Directional Dynamics



Introduction




Phugoid modeShort Period mode

Lateral-Directional DynamicsSpiral modeRoll modeDutch Roll mode



Introduction


Characteristic Equation

Taking Laplace transform of linearized governing equations

As4 + Bs3 + Cs2 + Ds + E = 0

Roots of characteristic equations ⇒ modesFor lateral-directional dynamics

(λ− λs)︸︷︷︸spiral

(λ− λr)︸︷︷︸roll

(λ2 + 2ζDωnDλ + ω2nD

)︸︷︷︸dutchroll=0



Introduction




As4 + Bs3 + Cs2 + Ds + E = 0

Roots of characteristic equations ⇒ modes

For lateral-directional dynamics







Introduction




As4 + Bs3 + Cs2 + Ds + E = 0

Roots of characteristic equations ⇒ modesFor lateral-directional dynamics







Introduction

Why an Analytical Solution

Numerical Solution

Characteristic equation is a polynomial equationCan be easily solved on a computer

does not give an insighthow different parameters affect various mode

Need for analytical solutionclassroom teachingaircraft designcontol law formulation



Introduction

Why an Approximate Solution

Analytical Solution

Characteristic equation is 4th degree polynomial equationExact solutions exist (Ferrari’s method)

complicated and lengthydoes not give an insight



Introduction


Approximate Solution

All terms in exact solution are not dominantOmitting non-dominant terms → approximate rootApproximation through meaningful physical assumptions



Introduction


Approximate Solution

All terms in exact solution is not dominantOmmiting non-dominat terms → approximate rootApproximation through meaningful physical assumptions

Example

Assumption of constant forward velocity gives an excellentapproximation to short period mode



Introduction

Scope

Research Oppurtunity

Longitudinal ModesGood approximations existed for short periodPhugoid approximations were looked into by manyresearchres

Lateral-Directional ModesStandard text books said good approximations existed forspiral and rollLiterature lacks good dutch roll approximations



New Approximations

Dutch Roll Frequency Approximation -

Derivation

As4 + Bs3 + Cs2 + Ds + E = (λ− λs)(λ− λr)(λ2 + 2ζDωnDλ + ω2nD )

Equating the coefficients

B

A= 2ζDωnD − λr − λs

C

A= ω2nD − 2ζDωnD(λr + λs) + λrλs

D

A= 2ζDωnDλrλs − (λr + λs)ω

2nD

E

A= ω2nDλrλs



New Approximations


Derivation



E

A= ω2nDλrλs



New Approximations


Derivation



E

A= ω2nDλrλs

A relation for dutch roll frequency is

ω2nD =(E/A)

λrλs



New Approximations


Derivation

ω2nD =(E/A)

λrλs

Dutch roll frequency depends on λr and λs

Derivation of an approximation for λs

Dλs + E = 0

Substituting this into expression for dutch roll frequencydutch roll frequency approximation heavily dependent onroll approximations



New Approximations


Derivation

ω2nD =(E/A)

λrλs

Dutch roll frequency depends on λr and λsDerivation of an approximation for λs

Aλ4s + Bλ3s + Cλ

2s + Dλs + E = 0




New Approximations


Derivation

ω2nD =(E/A)

λrλs


Dλs + E = 0




New Approximations


Derivation

ω2nD =(E/A)

λrλs


Dλs + E = 0

λs =−ED




New Approximations


Derivation

ω2nD =(E/A)

λrλs


Dλs + E = 0

λs =−ED

Substituting this into expression for dutch roll frequency

dutch roll frequency approximation heavily dependent onroll approximations



New Approximations


Derivation

ω2nD ≈−(D/A)

λr


Dλs + E = 0

λs =−ED

Substituting this into expression for dutch roll frequency

dutch roll frequency approximation heavily dependent onroll approximations



New Approximations


Derivation

ω2nD ≈−(D/A)

λr


Dλs + E = 0

λs =−ED




Roll Approximations

A Relook at Roll Approximations

Need to test the accuracy of existing roll approximationsA check over one or two cases doesn’t give confidenceA wide spectrum database is requiredDatabase given by Roskam

6 different airplanesin a total of 16 flight conditions



Roll Approximations

The Airplane Database

Aircraft Representative of:Cessna 172

A small, single piston enginegeneral aviation airplaneBeech M99

B small, twin turbopropregional commuter airplaneSIAI–Marchetti S211

C small, single jet enginemilitary training airplaneGates Learjet M24

D twin jet enginecorporate airplaneMcDonnell Douglas F4C

E twin jet enginefighter/attack airplaneBoeing 747

F large, four jet enginecommercial transport airplane



Roll Approximations









Flight Condition1 Low Altitude Cruise



Roll Approximations









Flight Condition1 Power Approach2 Low Altitude Cruise3 High Altitude Cruise



Roll Approximations









Flight Condition1 Power Approach2 Normal Cruise3 High Altitude Cruise



Roll Approximations









Flight Condition1 Power Approach2 Maximum Weight Cruise3 Low Weight Cruise



Roll Approximations









Flight Condition1 Power Approach2 Subsonic Cruise3 Supersonic Cruise



Roll Approximations









Flight Condition1 Power Approach2 High Altitude Cruise3 Low Altitude Cruise



Roll Approximations

Looking at Existing Roll Approximations

Test of Accuracy

%Error =ExactValue− ApproximateValue

ExactValue× 100

No simple yet, accurate and consistent approximationsexistNeed a new approximation



A New Roll Approximation

A New Roll Approximation: Development

Approximate Factorization

[λ2 +

(−Nr −

YβU1

+Lβg cos Θ1/U1

L2p + Nβ

)λ +

(L2p + Nβ +

YβU1

Nr

)]×[

λ2 +

(−Lp −

Lβg cos Θ1/U1L2p + Nβ

)λ +

(L2p −

LβNrg cos Θ1/U1L2p + Nβ

)]≈ 0

R. B. Russel, Performance and Stability of Aircraft,Butterworth-Heinemann, 1996.






[λ2 +

(−Nr −

YβU1

+Lβg cos Θ1/U1

L2p + Nβ

)λ +

(L2p + Nβ +

YβU1

Nr

)]×[

λ2 +

(−Lp −


)λ +

(L2p −


)]≈ 0

Actual Factorization[λ2 + 2ζDωnDλ + ω

2nD

][(λ− λr)(λ− λs)] = 0






[λ2 +

(−Nr −

YβU1

+Lβg cos Θ1/U1

L2p + Nβ

)λ +

(L2p + Nβ +

YβU1

Nr

)]×[

λ2 +

(−Lp −


)λ +

(L2p −


)]≈ 0

Actual Factorization

[λ2 + 2ζDωnDλ + ω

2nD

] [λ2 + (− λr − λs)λ + λrλs

]= 0




Accuracy of New Roll Approximation

The new roll approximation is

−Lp −Lβg cos Θ1/U1

L2p+Nβ

This approximation is not so accurateslips too much in some cases




Investigation of Roll

Analysis by visualizationA simulation/visualization package was developedRoll mode of each test case was carefully studied




The Simulation Package




Models for Visualization




Demonstration

Demonstration of the Simulation Package




Results of Analysis

Observation

1 Roll mode in most of cases studied involved pure roll2 But in some cases there was significant participation of yaw

and sideslip

Inference

A good roll mode approximation should respect theparticipation of yaw and sideslip

Implecation

Sideforce derivative Yβ and cross coupling derivatives Lr andNp should find respectable positions in the roll approximation




The New Roll Approximation

λr = L′p +

L′β

(g

U1−N ′p

)(L′2p + N

′β)

+Yβ(L

′rN

′p − L′pN ′r)

(L′2p + N′β)

Aircraft Flight New ApproxCondition % Error

A 1 −0.431 −0.94

B 2 0.203 −2.431 −5.21

C 2 0.423 0.281 −10.62

D 2 0.103 −0.941 1.70

E 2 −0.323 −0.111 −1.75

F 2 2.403 0.85




The New Roll Approximation

λr = L′p +

L′β

(g

U1−N ′p

)(L′2p + N

′β)

+Yβ(L

′rN

′p − L′pN ′r)

(L′2p + N′β)

Aircraft Flight New Approx TraditionalCondition % Error Approx.

A 1 −0.43 0.201 −0.94 14.84

B 2 0.20 4.563 −2.43 15.361 −5.21 7.24

C 2 0.42 1.443 0.28 3.001 −10.62 49.61

D 2 0.10 15.093 −0.94 11.811 1.70 3.55

E 2 −0.32 8.043 −0.11 −1.021 −1.75 13.60

F 2 2.40 6.193 0.85 10.24



Dutch Roll Approximation

The New Dutch Roll Frequency

Approximation

ω2nD =−(D/A)

λr

ω2nD =− (D/A)

L′p +

L′β(g

U1−N ′p)

(L′2p + N′β)

+Yβ(L

′rN

′p − L′pN ′r)

(L′2p + N′β)





Approximation

ω2nD =−(D/A)

λr

ω2nD =− (D/A)

L′p +

L′β(g

U1−N ′p)

(L′2p + N′β)

+Yβ(L

′rN

′p − L′pN ′r)

(L′2p + N′β)

D

A= −

Yβ

U1(L′pN

′r − L′rN ′p) +

Yp

U1(L′βN

′r −N ′βL

′r)−

g

U1cos Θ1L

′β

+ (L′βN′p −N ′βL

′p)−

Yr

U1(L′βN

′p −N ′βL

′p)





Approximation

ω2nD =−(D/A)

λr

ω2nD =− (D/A)

L′p +

L′β(g

U1−N ′p)

(L′2p + N′β)

+Yβ(L

′rN

′p − L′pN ′r)

(L′2p + N′β)

New Dutch Roll Frequency Approximation

ω2nD

=

g

U1L

′β + N

′βL

′p − L

′βN

′p

L′p +

L′β (g

U1− N′p)

(L′2p + N′β

)+

Yβ (L′rN

′p − L

′pN

′r)

(L′2p + N′β

)




Accuracy of New Dutch Roll Frequency

Approximation

Aircraft Flight New Approx.Condition % Error

A 1 1.132 −0.61

B 2 1.563 2.691 11.46

C 2 0.463 0.611 13.51

D 2 −0.173 1.131 −3.42

E 2 −0.323 −0.041 0.62

F 2 −0.253 −0.79




Accuracy of New Dutch Roll Frequency

Approximation

Aircraft Flight New Approx. TraditionalCondition % Error Approx.

A 1 1.13 10.352 −0.61 4.33

B 2 1.56 −4.673 2.69 −12.121 11.46 33.12

C 2 0.46 −14.163 0.61 −11.581 13.51 18.51

D 2 −0.17 −0.373 1.13 −1.131 −3.42 38.78

E 2 −0.32 9.083 −0.04 4.991 0.62 48.30

F 2 −0.25 1.363 −0.79 7.23



High Angle of Attack Flight Dynamics


one or more modes go unstable ⇒ DepartureLateral-Directional Departure

Wing Rockoscillations in roll

Nose Slicedivergence in yaw

Wing Dropdivergence in roll




Handling Quality at High Angle of Attack

spiral and roll modes combine to form a single oscillatorymode → ‘lateral phugoid’undesirable from handling qualities point of viewacceptable at certain flight phases with a speculation ondamping

2ζsrωnsr > k

Flight Phase Level Desired kCategory B & C 1 1Category B & C 2 0.6Category B & C 3 0.3




Proposal of a Flying Quality Criterion

2ζsrωnsr is equivalent to −λr − λsApproximation developed for roll should work well for2ζsrωnsrProposed approximation,

2ζsrωsr ≈ −L′p +L′β

N ′β

„N ′p −

g

U1

«

Accuracy evaluated using F16 database




The F16 Database

Aerodynamic coefficients were available in form of look-uptablesCoefficients as functions of α and βStability derivatives calculated as local slopes @ steadystateNeed knowledge of steady state velocity to calculatedimensional derivatives ⇒ Trim the aircraft

Nonlinear equations of motion were solved in anoptimization framework

For steady state, ẋ = f(x, t) = 0Cost function

Pi ẋ

2i = 0




Accuracy of New Flying Quality Criterion

2ζsrωsr ≈ −L′p +L′βN ′β

„N ′p −

g

U1

«

20 25 30 35 40 45

−0.2

0

0.2

0.4

0.6

α

2 ζ s

r ωsr

ExactProposed Approx.




A Recommendation

−L′p +L′βN ′β

(N ′p −

g

U1

)> k

Guidelines for aircraft preliminary designLongitudinal stability

10% static margin

Directional stabilityCnβ > 0.001 per deg

Lateral stabilitydoes not exista criterion on Clβ ?

L′β >

(k + L′p

)(N ′p −

gU1

)N ′βJoel George Lateral Directional Approximations to Aircraft


Conclusions

Conclusions

Highlightsan exhaustive review of existing lateral-directionalapproximationsderivation of a simple and accurate roll approximationa good approximation for dutch roll frequency

Future Workliterature lacks a good dutch roll damping approximationgood lateral-directional departure criteria is yet to comehandling quality criteria at high angle of attack is a nicepiece of research to take up


IntroductionAircraft Equations of MotionWhy an Analytical SolutionWhy an Approximate SolutionScope

New ApproximationsRoll ApproximationsRoll ApproximationsA New Roll ApproximationDutch Roll ApproximationHigh Angle of Attack Flight DynamicsConclusions

Documents

Lateral Directional Approximations to Aircraft · 2013-02-19 · Lateral Directional Approximations to Aircraft Lateral Directional Approximations to Aircraft Joel George Department