М.Г.Гоман «Численный анализ нелинейной динамики систем», доклад на 1-й конференции Института математики и приложений (IMA) по фрактальной геометрии, г.Лейстер (Великобритания), 19 сентября 2000 года. M.G.Goman "Computational Analysis of Nonlinear Dynamical Systems ", presentation at the IMA (Institute of Mathematics and its Applications) 1st Conference in Fractal Geometry, De Montfort University, Leicester, the UK, 19 September 2000.

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Computational Analysis of Nonlinear
Dynamical Systems
M.G.Goman
Institute 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 aircraft
Enhanced Fighter Maneuverability (EFM) demonstrator
The Herbst Maneuver
V
a=70
T
Thrust 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-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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Delay in Asymmetry Onset
Wind Tunnel Experiment

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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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Asymmetrical Vortex Breakdown
Water Tunnel Experiment

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Vortex Breakdown Hysteresis
Water Tunnel Experiment

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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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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 w
0.00
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 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
( )