М.Г.Гоман, А.В.Храмцовский (2000) - Численный анализ...

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

ICNPAA, Florida, 10-12 May, 2000 3

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

ICNPAA, Florida, 10-12 May, 2000 6

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

ICNPAA, Florida, 10-12 May, 2000 7

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

ICNPAA, Florida, 10-12 May, 2000 8

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

ICNPAA, Florida, 10-12 May, 2000 9

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

ICNPAA, Florida, 10-12 May, 2000 10

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

ICNPAA, Florida, 10-12 May, 2000 11

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

ICNPAA, Florida, 10-12 May, 2000 12

KRIT GUI for Post-Processing of Continuation Database

ICNPAA, Florida, 10-12 May, 2000 13

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