ICNPAA, Florida, 10-12 May, 2000 1

Computer Aided Analysis of Aircraft Nonlinear Dynamics

M.G.Goman and A.V.KhramtsovskyCentral Aerohydrodynamic Institute (TsAGI), Russia

De Montfort University, UK

ICNPAA, Florida, 10-12 May, 2000 2

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