М.Г.Гоман (2000 final) – Численный анализ нелинейной...
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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.TRANSCRIPT
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21 September 2000 IMA 1st Conference in Fractal Geometry 1
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( )