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Computational Analysis of NonlinearDynamical Systems

M.G.GomanInstitute of Mathematical and Simulation Scineces

De Montfort University, UK

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Contents:

• Nonlinear Dynamics Problems from Aeronautics- Multi-Attractor Aircraft Dynamics (computational study)

- Aerodynamic Asymmetry at High Incidence (experimental results interpretation)

• KRIT Toolbox for Nonlinear Investigation and Examples of its Application

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Expanding the Frontiers of Flight

• Design objectives: - stealth, high incidence and agility, larger scale and lighter structure, active control approach, etc.

• Increasing role of mathematical modelling in design process

• Integrated, coupled and nonlinear mathematical models

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The F/A-18A Hornet HARV

Vortex core

Vortex breakdown

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The X31 aircraftEnhanced Fighter Maneuverability (EFM) demonstrator

The Herbst Maneuver

V

a=70

TThrust vectoring

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Wind Tunnel Tests

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Beyond the Normal Flight

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Aircraft Rigid Body Dynamics

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KRIT Toolbox for Nonlinear Dynamics Analysis

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Multi-Attractor Dynamics Investigation (I)

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Multi-Attractor Dynamics Investigation (II)

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KRIT GUI for Phase Portrait Design

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KRIT GUI for Continuation

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KRIT GUI for Numerical Simulation

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Aerodynamic Asymmetry for X-31Flight Tests

Data range

Unmodified forebodyForebody and noseboomtransition strip

Data range

-.10 -.05 0 .05 .10 -.10 -.05 0 .05 .1020

30

40

50

60

70

80

Ang

le o

f atta

ck (d

eg)

C Cn0 n0

Trust vectoringRudder

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Delay in Asymmetry OnsetWind Tunnel Experiment

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Onset of Vortical Flow AsymmetrySimplified math model

1020

1 2-1-2

C /l e

b/e

2

a/e acr

C > 0l

C = 0l

C < 0l

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Asymmetrical Vortex BreakdownWater Tunnel Experiment

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Vortex Breakdown HysteresisWater Tunnel Experiment

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Asymmetry Onset with Vortex BreakdownWater Tunnel Experiment

x x1 21

0.5

0

-0.5

-1.0

Asymmetry of vortex breakdown points at zero sideslip

- "noisy" tunnel- "quiet" tunnel

1 2X=X -X Supposed structureof vortex breakdownsteady states

chaotic behaviour

20 25 30 40 45Angle of attack (deg)

35

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Stability of multiple state vortical flow at the presence of external disturbances

("potential function" analogy)

- level of disturbances

a)

a)

b)

b)

c)

transmitting chaotic behaviour

disturbed stable flow

disturbed bistable flow

c)

supercritical bifurcation

Clav=0

delay of asymmetry onset

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Double-Well Dynamical System

0.6 0.7 0.8 0.9 w0.00

0.04

0.08

0.12

f

Periodical predictable dynamics

Fractal stability boundaries

Chaotic dynamics

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Conclusion Remarks:

• The KRIT Toolbox in Matlab provides a broad range of numerical procedures and graphical user interfaces (GUI) for: - nonlinear aircraft dynamics investigation,- post-design control laws assessment - assistance in piloted simulation

• The Toolbox for general nonlinear dynamics problems is under development

• The work during last several years was funded by DERA, UK

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Hysteresis in Steady Vortex BreakdownWater Tunnel Experiment

Supposed structureof vortex breakdownsteady states

Asymmetrychange

"noisy" tunnel"noisy" tunnel

"quiet" tunnel

- a=35- a=40- a=40

0 1 2 3 4 5 6-1-2-3-4-5-6-7 7

Sideslip (deg)

Asymmetry of vortex breakdown at sideslip

00.1

0.2

0.30.4

0.50.60.70.8

-0.1

-0.2-0.3

-0.4-0.5-0.6-0.7

-0.8

X=X -X1 2( )