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Computer Aided Analysis of Aircraft Nonlinear Dynamics
M.G.Goman and A.V.KhramtsovskyCentral Aerohydrodynamic Institute (TsAGI), Russia
De Montfort University, UK
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Contents:
• Nonlinear Aircraft Dynamics Problems• Qualitative Analysis of Multi-Attractor Dynamics• Numerical Methods for Qualitative Analysis• KRIT Toolbox for Nonlinear Dynamics Problems• Examples of the KRIT Application
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Nonlinear Aircraft Dynamics Problems
• Critical flight regimes investigation at high incidence and intensive maneuvering (roll-coupling, wing rock, spin, etc.) Objectives: departure prevention and recovery design.
• Closed-loop system dynamics analysis. - Post-design control laws assessment.
• Assistance in piloted simulation. - Pilot training at high incidence flight.
- Aircraft
Controller
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Beyond the Normal Flight
• Critical flight regimes
• Supermaneuverability arena
• Multiple-attractor dynamics
25-30 50-60
Flat Spin
Steep Spin
Roll Coupling:
Angle of Attack
Velo
city
Rol
l Rat
e,
autorotation regimes
Deep stall regimesDepartures:
wing rock, nose slice,etc.
Normal FlightRegimes
a, deg
w
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High Angle of Attack Flight Dynamics Problems
FlightTests
Simulation&
Stability&
DynamicsAnalysis
ControlLaws
Design
AerodynamicsModelling Pilot Training
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Aircraft Rigid Body Dynamics
Equations of Motion
State Variables
Control Variables
w
w
w
ww
d
d
d
d
dd
d
d
t
t
t
tI + Ix = Ma+Mc
Vm + V = F+ T + Gx( ) a
R =
=
C( )
)
VQ
QQ E(
Q q f y
w
a
a
bb
b
d d d d d d
h h z
R ==
=
(X Y Z )g g g
(
(
)
)
V
= p q r T
T
T
T
T
VVV
cos cos
cossinsin
e er l
l l
=
=
(
(
)
)
a r c...
T T Tr r
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KRIT Toolbox for Nonlinear Aircraft Dynamics Analysis
= F( ,x x c. ) ,
Equilibrium statesF( ,x c) = 0
:
systematic search methodlocal stability analysis
Closed orbits:jt(x ) = x* *
Poincare mapping techniquemultipliers analysis
computation of two dimensionalcross section of regions of attraction
Stability "in large" analysisGlobal dynamics analysis
of multiple attractors
Continuation and bifurcation analysis
Rn
Rn
RmX
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Qualitative Analysis of Multi-Attractor Dynamics
• Phase portrait design at fixed controls and parameters - equilibrium and periodic solutions,local stability characteristics- reconstruction of attraction regions
• Continuation of equilibria, closed orbits, etc. with controls/parameters
• Bifurcation analysis and departure prediction
• Inspecting numerical simulation
stable point
stable point
stable point
saddle point
saddle point
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Critical Elements of Phase Portrait
• Stable elements (attractors): - steady normal flight (equilibria); - critical flight regimes: wing-rock, spin, roll-coupled inertia rotation (equilibria, closed-orbits, toroidalmanifolds, chaotic attractor)
• Stable and unstable manifolds of trajectories for unstable elements (repellers): - boundaries of attraction regions - topological link between different elements
Equilibrium point Closed orbit
Toroidal manifold Chaotic attractor
W
W
W
sn-1
n-1
u
u
1
1
L
W
W
u2
sn-1
Gn-1,2
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Bifurcation Analysis of Equilibrium States and Closed Orbits
Re
Re
l
l
l
l
Im
Im
0
i
Fxdet 0
Saddle-node bifurcation
limit point
transcritical casesubcritical casesupercritical case
branchin point
Andronov-Hopf bifurcation
supercritical
subcritical
Equilibrium States
Im ImImIm
r rrr
r rrr
1 Re ReReRe -1
ee iijj
ee-i-ijj
G n-1,2 G n-2,3 G
GT
2T
T n,2
Closed Orbits
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Numerical Methods for Qualitative Analysis
• Continuation algorithm: - branching and ‘kink’ points processing; - systematic search for all solutions of nonlinear system at fixed parameters; - bifurcation points identification and collection
• Regions of attraction:- reconstruction of stability region boundary; - computation of two-dimensional cross sections
• Numerical simulation: - perturbations in particular manifolds of trajectories;
detF = 0
det = 0Fx
s
x
c
limit point
branching point
parameter variation
guaranteed estimate ofdomain of attraction forequilibrium point
two-dimensional cross section P2
stable manifold of trajectories W ofsaddle equilibrium point
n-1s
guaranteed estimate ofdomain of attraction forclosed orbit fixed point
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KRIT GUI for Post-Processing of Continuation Database
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KRIT GUI for Phase Portrait Design
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KRIT GUI for Numerical Simulation
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Conclusions:
• The KRIT Toolbox in Matlab provides a broad range of numerical procedures and graphical user interfaces (GUI) for: - nonlinear aircraft dynamics investigation at high angles of attack, - post-design control laws assessment and - assistance of piloted simulation at high incidence flight
• The work was funded during last several years by Defence Evaluation and Research Agency of MoD, UK