FMFP 2016
Investigating a Fluid-Elastic Strange Attractor in Dipteran Wing Motor Inspired Flapping Motion
Chandan Bose1, Vikas Yettella2, Chakshu Deora2, Sunetra Sarkar2 1Department
of Applied Mechanics, 2Department of Aerospace Engineering Indian Institute of Technology, Madras
15.12.2016
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MOTIVATION
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Biomimicry: Inspiration from Nature
Click-compliant mechanism for flapping wing MAV
Source: Wikipedia
Source: Chin et al.,2012
How do insects fly? Ref: Chin et al., IEEE Proceedings, 2012
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Role of Insect Flight Motor
Dipteran insect’s flight thorax
Ref: Hendenstrom, Plos Biology, 2014 Ref: Lau et al., IEEE Transactions Robotics, 2014
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MODELING
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Modeling Insect Flight Motor
Ref: Brennan et al., J. Theo. Bio, 2003 Ref: Harne and Wang, J. Interface of Royal Society, 2015
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COMPUTATIONAL METHODOLOGY
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Computational Methodology • Two separate methodologies have been used for the structural and flow part in the present FSI model.
• A forced Duffing oscillator model – for the structural part • Lumped Vortex Method – for evaluating the fluid load • Both the solvers have been integrated in a staggered form to simulate the FSI response.
A partitioned based loose coupling method 8
Structural Model: Forced Duffing Oscillator
1 1 3 L( ) u u u f nd sin( ) 2 2 2 0 mD ..
Ref: Brennan et al., J. Theo. Bio, 2003
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Flow Solver: Lumped Vortex Method • Flow is inviscid. • Thickness of the body is negligible. • Body deformations are small.
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The flat plate is divided into N panels of length a.
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One vortex element of strength of is placed at control point of each panel (the quarter chord point). The no normal boundary condition is to be satisfied at the collocation point of each panel (the 3/4th chord point).
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Ref: J. Katz & A. Plotkin, Cambridge University Press, 2001
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Discrete Wake Model •
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At every time step one vortex element leaves the plate from the trailing edge and goes to the wake. Wake vortices move with a velocity induced by the free stream, body bound vortices and the other free vortices remaining in the wake N boundary conditions at N collocation points gives N equations. To calculate the strength of the wake vortex leaving the trailing edge one more boundary condition is needed. Other boundary condition comes from Kelvin circulation theorem as:
Figure Source: J. Katz & A. Plotkin, Low-speed aerodynamics, Cambridge University Press 13(2001)
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Load Calculation • The pressure difference between the top and bottom surface of each panel can be obtained using the unsteady Bernoulli equation as
• Lift force acting on the
panel is given by,
• Total lift acting on the flat plate is,
• This aerodynamic lift acts on the structure along with the wing actuation force. 12
Flow Solver Validation
Ref: Young, J., PhD Thesis, The University of New South Wales, 2005
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Periodic Regime A double well attractor
A = Large amplitude oscillation
B,C = Small amplitude oscillation 14
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RESULTS & DISCUSSIONS
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Small Amplitude Oscillation
fnd=0.004
fnd=0.02
fnd=0.146
fnd=0.14
fnd=0.148
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fnd=0.228
fnd=0.20
fnd=0.212
Large Amplitude Oscillation
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Transient Chaos (fnd=0.16)
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Transient Chaos (fnd=0.16)
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Transient Chaos (fnd=0.16)
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Sustained Chaos (fnd=0.24)
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Sustained Chaos (fnd=0.24)
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Strange attractor
fnd=0.24 23
Largest Lyapunov Exponent
LLE = 0.82
• Rosensteine Algorithm Ref: Rosenstein et al., Physica D, 1993
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Periodic Flow Field
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Chaotic Flow Field
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Limitation and Future Direction Inviscid & small displacement
Use of Navier-Stokes solver
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CONCLUDING REMARKS
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Summary •
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The transition in the nonlinear FSI dynamics of a flexible insect flight motor has been investigated modeling it as a forced Duffing oscillator in the presence of an inviscid fluid. The nonlinear FSI model has been developed using a lumped vortex method based potential flow solver coupled with nonlinear structural model by partitioned approach based weak coupling method. A bifurcation analysis has been performed considering the actuation force amplitude as the control parameter. The flapping response transitions from a periodic to chaotic dynamics through a periodic doubling route. The aerodynamic loads as well as the wake pattern also follows the same route. The periodic attractors undergo a doubling cascade giving birth to a strange chaotic attractor reflecting the complex flight patterns. An interesting transient chaotic state has been observed much prior to reaching stable chaotic regime. The strange attractor in the Poincaŕe section and a positive LLE confirms the existence of the chaotic attractor. 29
Thank you!
Contact:
[email protected],
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