Lateral Directional Approximations to Aircraft · 2013-02-19 · Lateral Directional Approximations...
Transcript of Lateral Directional Approximations to Aircraft · 2013-02-19 · Lateral Directional Approximations...
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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
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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
Joel George Lateral Directional Approximations to Aircraft
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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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
Introduction
Aircraft Equations of Motion
6 Dof Equations of Motion
12 nonlinear 1st order ODE6 Dynamic Equations
3 Force Equations3 Moment Equations
6 Kinematic Equations
Linearize the equations about an equillibrium point
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
Introduction
Aircraft Equations of Motion
6 Dof Equations of Motion
12 nonlinear 1st order ODE6 Dynamic Equations
3 Force Equations3 Moment Equations
6 Kinematic Equations3 Euler Angle Equations3 Navigation Equations
Linearize the equations about an equillibrium point
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
Introduction
Aircraft Equations of Motion
6 Dof Equations of Motion
12 nonlinear 1st order ODE6 Dynamic Equations
3 Force Equations3 Moment Equations
6 Kinematic Equations3 Euler Angle Equations3 Navigation Equations
Linearize the equations about an equillibrium point
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
Introduction
Aircraft Equations of Motion
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
Introduction
Aircraft Equations of Motion
Decoupling of Dynamics
Linearized aircraft dynamics can be decoupled asLongitudinal Dynamics
Phugoid modeShort Period mode
Lateral-Directional Dynamics
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
Introduction
Aircraft Equations of Motion
Decoupling of Dynamics
Linearized aircraft dynamics can be decoupled asLongitudinal Dynamics
Phugoid modeShort Period mode
Lateral-Directional DynamicsSpiral modeRoll modeDutch Roll mode
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
Introduction
Aircraft Equations of Motion
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
Introduction
Aircraft Equations of Motion
Characteristic Equation
Taking Laplace transform of linearized governing equations
As4 + Bs3 + Cs2 + Ds + E = 0
Roots of characteristic equations ⇒ modes
For lateral-directional dynamics
(λ− λs)︸ ︷︷ ︸spiral
(λ− λr)︸ ︷︷ ︸roll
(λ2 + 2ζDωnDλ + ω2nD
)︸ ︷︷ ︸dutchroll=0
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
Introduction
Aircraft Equations of Motion
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
Introduction
Why an Approximate Solution
Approximate Solution
All terms in exact solution are not dominantOmitting non-dominant terms → approximate rootApproximation through meaningful physical assumptions
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
Introduction
Why an Approximate Solution
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
New Approximations
Dutch Roll Frequency Approximation -
Derivation
As4 + Bs3 + Cs2 + Ds + E = (λ− λs)(λ− λr)(λ2 + 2ζDωnDλ + ω2nD )
Equating the coefficients
E
A= ω2nDλrλs
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
New Approximations
Dutch Roll Frequency Approximation -
Derivation
As4 + Bs3 + Cs2 + Ds + E = (λ− λs)(λ− λr)(λ2 + 2ζDωnDλ + ω2nD )
Equating the coefficients
E
A= ω2nDλrλs
A relation for dutch roll frequency is
ω2nD =(E/A)
λrλs
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
New Approximations
Dutch Roll Frequency Approximation -
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
New Approximations
Dutch Roll Frequency Approximation -
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
Substituting this into expression for dutch roll frequencydutch roll frequency approximation heavily dependent onroll approximations
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
New Approximations
Dutch Roll Frequency Approximation -
Derivation
ω2nD =(E/A)
λrλs
Dutch roll frequency depends on λr and λsDerivation of an approximation for λs
Dλs + E = 0
Substituting this into expression for dutch roll frequencydutch roll frequency approximation heavily dependent onroll approximations
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
New Approximations
Dutch Roll Frequency Approximation -
Derivation
ω2nD =(E/A)
λrλs
Dutch roll frequency depends on λr and λsDerivation of an approximation for λs
Dλs + E = 0
λs =−ED
Substituting this into expression for dutch roll frequencydutch roll frequency approximation heavily dependent onroll approximations
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
New Approximations
Dutch Roll Frequency Approximation -
Derivation
ω2nD =(E/A)
λrλs
Dutch roll frequency depends on λr and λsDerivation of an approximation for λs
Dλs + E = 0
λs =−ED
Substituting this into expression for dutch roll frequency
dutch roll frequency approximation heavily dependent onroll approximations
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
New Approximations
Dutch Roll Frequency Approximation -
Derivation
ω2nD ≈−(D/A)
λr
Dutch roll frequency depends on λr and λsDerivation of an approximation for λs
Dλs + E = 0
λs =−ED
Substituting this into expression for dutch roll frequency
dutch roll frequency approximation heavily dependent onroll approximations
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
New Approximations
Dutch Roll Frequency Approximation -
Derivation
ω2nD ≈−(D/A)
λr
Dutch roll frequency depends on λr and λsDerivation of an approximation for λs
Dλs + E = 0
λs =−ED
Substituting this into expression for dutch roll frequencydutch roll frequency approximation heavily dependent onroll approximations
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Flight Condition1 Low Altitude Cruise
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Flight Condition1 Power Approach2 Low Altitude Cruise3 High Altitude Cruise
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Flight Condition1 Power Approach2 Normal Cruise3 High Altitude Cruise
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Flight Condition1 Power Approach2 Maximum Weight Cruise3 Low Weight Cruise
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Flight Condition1 Power Approach2 Subsonic Cruise3 Supersonic Cruise
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Flight Condition1 Power Approach2 High Altitude Cruise3 Low Altitude Cruise
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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.
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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.
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
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
Actual Factorization[λ2 + 2ζDωnDλ + ω
2nD
][(λ− λr)(λ− λs)] = 0
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
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
Actual Factorization[λ2 + 2ζDωnDλ + ω
2nD
][(λ− λr)(λ− λs)] = 0
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
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
Actual Factorization
[λ2 + 2ζDωnDλ + ω
2nD
] [λ2 + (− λr − λs)λ + λrλs
]= 0
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
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
Actual Factorization
[λ2 + 2ζDωnDλ + ω
2nD
] [λ2 + (− λr − λs)λ + λrλs
]= 0
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
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
Actual Factorization
[λ2 + 2ζDωnDλ + ω
2nD
] [λ2 + (− λr − λs)λ + λrλs
]= 0
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
A New Roll Approximation
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
A New Roll Approximation
Investigation of Roll
Analysis by visualizationA simulation/visualization package was developedRoll mode of each test case was carefully studied
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
A New Roll Approximation
The Simulation Package
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
A New Roll Approximation
Models for Visualization
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
A New Roll Approximation
Demonstration
Demonstration of the Simulation Package
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
A New Roll Approximation
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
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
A New Roll Approximation
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
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
A New Roll Approximation
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
A New 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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
A New 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 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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
A New 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 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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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)
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
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)
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)
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
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)
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)
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
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)
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
)
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
Dutch Roll Approximation
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
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
Dutch Roll Approximation
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
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
Dutch Roll Approximation
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
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
High Angle of Attack Flight Dynamics
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
High Angle of Attack Flight Dynamics
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
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
High Angle of Attack Flight Dynamics
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
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
High Angle of Attack Flight Dynamics
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
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
High Angle of Attack Flight Dynamics
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
Joel George Lateral Directional Approximations to Aircraft
-
Lateral Directional Approximations to Aircraft
High Angle of Attack Flight Dynamics
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.
Joel George Lateral Directional Approximations to Aircraft
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Lateral Directional Approximations to Aircraft
High Angle of Attack Flight Dynamics
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
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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
Joel George Lateral Directional Approximations to Aircraft
IntroductionAircraft Equations of MotionWhy an Analytical SolutionWhy an Approximate SolutionScope
New ApproximationsRoll ApproximationsRoll ApproximationsA New Roll ApproximationDutch Roll ApproximationHigh Angle of Attack Flight DynamicsConclusions