The document outlines the proceedings of the 1st conference on fractal geometry held on September 21, 2000, focusing on nonlinear dynamics in aeronautics, including aircraft dynamics and the application of mathematical modeling in design. It discusses various investigations into multi-attractor dynamics, vortex behavior, and the use of the K.R.I.T. toolbox for analyzing nonlinear systems. The conclusion emphasizes the toolbox's utility for investigating aircraft dynamics and its ongoing development for general nonlinear problems.

1. 21 September 2000
IMA 1st Conference in Fractal
Geometry
1
Computational Analysis of Nonlinear
Dynamical Systems
M.G.Goman
Institute of Mathematical and Simulation Scineces
De Montfort University, UK

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IMA 1st Conference in Fractal
Geometry
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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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IMA 1st Conference in Fractal
Geometry
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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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IMA 1st Conference in Fractal
Geometry
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The F/A-18A Hornet HARV
Vortex core
Vortex breakdown

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IMA 1st Conference in Fractal
Geometry
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The X31 aircraft
Enhanced Fighter Maneuverability (EFM) demonstrator
The Herbst Maneuver
V
a=70
T
Thrust vectoring

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IMA 1st Conference in Fractal
Geometry
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Wind Tunnel Tests

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IMA 1st Conference in Fractal
Geometry
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Beyond the Normal Flight

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Geometry
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Aircraft Rigid Body Dynamics

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IMA 1st Conference in Fractal
Geometry
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KRIT Toolbox for Nonlinear Dynamics Analysis

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IMA 1st Conference in Fractal
Geometry
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Multi-Attractor Dynamics Investigation (I)

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IMA 1st Conference in Fractal
Geometry
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Multi-Attractor Dynamics Investigation (II)

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Geometry
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Geometry
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Geometry
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Geometry
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Geometry
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KRIT GUI for Phase Portrait Design

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Geometry
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KRIT GUI for Continuation

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IMA 1st Conference in Fractal
Geometry
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KRIT GUI for Numerical Simulation

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Geometry
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Aerodynamic Asymmetry for X-31
Flight Tests
Data range
Unmodified forebody
Forebody and noseboom
transition strip
Data range
-.10 -.05 0 .05 .10 -.10 -.05 0 .05 .10
20
30
40
50
60
70
80
Angleofattack(deg)
C C
n0 n0
Trust vectoring
Rudder

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IMA 1st Conference in Fractal
Geometry
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Delay in Asymmetry Onset
Wind Tunnel Experiment

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IMA 1st Conference in Fractal
Geometry
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Onset of Vortical Flow Asymmetry
Simplified math model
10
20
1 2-1-2
C /l e
b/e
2
a/e
acr
C > 0l
C = 0l
C < 0l

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IMA 1st Conference in Fractal
Geometry
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Asymmetrical Vortex Breakdown
Water Tunnel Experiment

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Geometry
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Vortex Breakdown Hysteresis
Water Tunnel Experiment

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Geometry
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Asymmetry Onset with Vortex Breakdown
Water Tunnel Experiment
x x1 2
1
0.5
0
-0.5
-1.0
Asymmetry of vortex breakdown points at zero sideslip
- "noisy" tunnel
- "quiet" tunnel
1 2X=X -X
Supposed structure
of vortex breakdown
steady states
chaotic behaviour
20 25 30 40 45
Angle of attack (deg)
35

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IMA 1st Conference in Fractal
Geometry
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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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IMA 1st Conference in Fractal
Geometry
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Double-Well Dynamical System
0.6 0.7 0.8 0.9 w
0.00
0.04
0.08
0.12
f
Periodical predictable dynamics
Fractal stability boundaries
Chaotic dynamics

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Geometry
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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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Geometry
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Geometry
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Hysteresis in Steady Vortex Breakdown
Water Tunnel Experiment
Supposed structure
of vortex breakdown
steady states
Asymmetry
change
"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
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
X=X -X1 2
( )