Research Papers

Transient and Stable Chaos in Dipteran Flight Inspired Flapping Motion

[+] Author and Article Information
Chandan Bose

Department of Applied Mechanics,
Indian Institute of Technology Madras,
Chennai 600036, India
e-mail: cb.ju.1991@gmail.com

Vikas Reddy

Department of Aerospace Engineering,
Indian Institute of Technology Madras,
Chennai 600036, India
e-mail: yettellavikas@gmail.com

Sayan Gupta

Department of Applied Mechanics,
Indian Institute of Technology Madras,
Chennai 600036, India
e-mail: gupta.sayan@gmail.com

Sunetra Sarkar

Department of Aerospace Engineering,
Indian Institute of Technology Madras,
Chennai 600036, India
e-mail: sunetra.sarkar@gmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 5, 2017; final manuscript received November 4, 2017; published online December 14, 2017. Assoc. Editor: Katrin Ellermann.

J. Comput. Nonlinear Dynam 13(2), 021014 (Dec 14, 2017) (9 pages) Paper No: CND-17-1359; doi: 10.1115/1.4038447 History: Received August 05, 2017; Revised November 04, 2017

This paper deals with the nonlinear fluid structure interaction (FSI) dynamics of a Dipteran flight motor inspired flapping system in an inviscid fluid. In the present study, the FSI effects are incorporated to an existing forced Duffing oscillator model to gain a clear understanding of the nonlinear dynamical behavior of the system in the presence of aerodynamic loads. The present FSI framework employs a potential flow solver to determine the aerodynamic loads and an explicit fourth-order Runge–Kutta scheme to solve the structural governing equations. A bifurcation analysis has been carried out considering the amplitude of the wing actuation force as the control parameter to investigate different complex states of the system. Interesting dynamical behavior including period doubling, chaotic transients, periodic windows, and finally an intermittent transition to stable chaotic attractor have been observed in the response with an increase in the bifurcation parameter. Similar dynamics is also reflected in the aerodynamic loads as well as in the trailing edge wake patterns.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Lepora, N. F. , Verschure, P. , and Prescott, T. J. , 2013, “ The State of the Art in Biomimetics,” Bioinspiration Biomimetics, 8(1), p. 013001. [CrossRef] [PubMed]
Ward, T. A. , Rezadad, M. , Fearday, C. J. , and Viyapuri, R. , 2015, “ A Review of Biomimetic Air Vehicle Research: 1984–2014,” Int. J. Micro Air Veh., 7(3), pp. 375–394. [CrossRef]
Ellington, C. P. , 1999, “ The Novel Aerodynamics of Insect Flight: Applications to Micro-Air Vehicles,” J. Exp. Biol., 202(23), pp. 3439–3448. http://jeb.biologists.org/content/202/23/3439.short [PubMed]
Alexander, R. M. , and Bennet-Clark, H. , 1977, “ Storage of Elastic Strain Energy in Muscle and Other Tissues,” Nature, 265(5590), pp. 114–117. [CrossRef] [PubMed]
Harne, R. , and Wang, K. , 2015, “ Dipteran Wing Motor-Inspired Flapping Flight Versatility and Effectiveness Enhancement,” J. R. Soc. Interface, 12(104), p. 20141367. [CrossRef] [PubMed]
Lau, G.-K. , Chin, Y.-W. , Goh, J. T.-W. , and Wood, R. J. , 2014, “ Dipteran-Insect-Inspired Thoracic Mechanism With Nonlinear Stiffness to Save Inertial Power of Flapping-Wing Flight,” IEEE Trans. Rob., 30(5), pp. 1187–1197. [CrossRef]
Chin, Y.-W. , and Lau, G.-K. , 2012, “ ‘Clicking’ Compliant Mechanism for Flapping-Wing Micro Aerial Vehicle,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Vilamoura, Portugal, Oct. 7–12, pp. 126–131.
Boettiger, E. G. , and Furshpan, E. , 1952, “ The Mechanics of Flight Movements in Diptera,” Biol. Bull., 102(3), pp. 200–211. [CrossRef]
Pringle, J. , 1957, Insect Flight, Cambridge University Press, Cambridge, UK.
Dickinson, M. H. , and Tu, M. S. , 1997, “ The Function of Dipteran Flight Muscle,” Comp. Biochem. Physiol., Part A: Mol. Integr. Physiol., 116(3), pp. 223–238. [CrossRef]
Thomson, A. J. , and Thompson, W. A. , 1977, “ Dynamics of a Bistable System: The Click Mechanism in Dipteran Flight,” Acta Biotheor., 26(1), pp. 19–29. [CrossRef]
Brennan, M. , Elliott, S. , Bonello, P. , and Vincent, J. , 2003, “ The ‘Click’ Mechanism in Dipteran Flight: If It Exists, Then What Effect Does It Have?,” J. Theor. Biol., 224(2), pp. 205–213. [CrossRef] [PubMed]
Tang, B. , and Brennan, M. , 2011, “ On the Dynamic Behaviour of the ‘Click’ Mechanism in Dipteran Flight,” J. Theor. Biol., 289, pp. 173–180. [CrossRef] [PubMed]
Pfau, H. K. , 1987, “ Critical Comments on a Novel Mechanical Model of Dipteran Flight,” J. Exp. Biol., 128(1), pp. 463–468. http://jeb.biologists.org/content/128/1/463
Miyan, J. A. , and Ewing, A. W. , 1985, “ Is the ‘Click’ Mechanism of Dipteran Flight an Artefact of CC14 Anaesthesia?,” J. Exp. Biol., 116(1), pp. 313–322. http://jeb.biologists.org/content/116/1/313
Miyan, J. A. , and Ewing, A. W. , 1985, “ How Diptera Move Their Wings: A Re-Examination of the Wing Base Articulation and Muscle Systems Concerned With Flight,” Philos. Trans. R. Soc. London, Ser. B, 311(1150), pp. 271–302. [CrossRef]
Chin, Y. W. , and Lau, G. K. , 2013, “ Is Clicking Mechanism Good for Flapping Wing Micro Aerial Vehicle?,” Proc. SPIE, 8686(86860W), p. 12. https://www.spiedigitallibrary.org/conference-proceedings-of-spie/8686/1/Is-clicking-mechanism-good-for-flapping-wing-micro-aerial-vehicle/10.1117/12.2009627.short?SSO=1
Kok, J. , Lau, G. , and Chahl, J. , 2016, “ On the Aerodynamic Efficiency of Insect-Inspired Micro Aircraft Employing Asymmetrical Flapping,” J. Aircr., 55(3), pp. 800–810. [CrossRef]
Hilborn, R. C. , 2000, Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers, Oxford University Press, Oxford, UK. [CrossRef]
Cao, Q. , Xiong, Y. , and Wiercigroch, M. , 2013, “ A Novel Model of Dipteran Flight Mechanism,” Int. J. Dyn. Control, 1(1), pp. 1–11. [CrossRef]
Katz, J. , and Plotkin, A. , 2001, Low-Speed Aerodynamics, Vol. 13, Cambridge University Press, Cambridge, UK. [CrossRef]
Fitzgerald, T. , Valdez, M. , Vanella, M. , Balaras, E. , and Balachandran, B. , 2011, “ Flexible Flapping Systems: Computational Investigations Into Fluid-Structure Interactions,” Aeronaut. J., 115(1172), pp. 593–604. [CrossRef]
Lai, Y.-C. , and Tél, T. , 2011, Transient Chaos: Complex Dynamics on Finite Time Scales, Vol. 173, Springer Science & Business Media, New York.
Grebogi, C. , Ott, E. , and Yorke, J. A. , 1983, “ Crises, Sudden Changes in Chaotic Attractors, and Transient Chaos,” Phys. D, 7(1–3), pp. 181–200. [CrossRef]
Leven, R. , Pompe, B. , Wilke, C. , and Koch, B. , 1985, “ Experiments on Periodic and Chaotic Motions of a Parametrically Forced Pendulum,” Phys. D, 16(3), pp. 371–384. [CrossRef]
Sommerer, J. C. , Ku, H.-C. , and Gilreath, H. E. , 1996, “ Experimental Evidence for Chaotic Scattering in a Fluid Wake,” Phys. Rev. Lett., 77(25), p. 5055. [CrossRef] [PubMed]
Tam, J. I. , and Holmes, P. , 2014, “ Revisiting a Magneto-Elastic Strange Attractor,” J. Sound Vib., 333(6), pp. 1767–1780. [CrossRef]
Wolf, A. , Swift, J. B. , Swinney, H. L. , and Vastano, J. A. , 1985, “ Determining Lyapunov Exponents From a Time Series,” Phys. D, 16(3), pp. 285–317. [CrossRef]
Eckmann, J.-P. , Kamphorst, S. O. , and Ruelle, D. , 1987, “ Recurrence Plots of Dynamical Systems,” Europhys. Lett., 4(9), p. 973. [CrossRef]
Marwan, N. , Romano, M. C. , Thiel, M. , and Kurths, J. , 2007, “ Recurrence Plots for the Analysis of Complex Systems,” Phys. Rep., 438(5), pp. 237–329. [CrossRef]
Grossmann, A. , Kronland-Martinet, R. , and Morlet, J. , 1990, “ Reading and Understanding Continuous Wavelet Transforms,” Wavelets, Springer, Berlin, pp. 2–20.


Grahic Jump Location
Fig. 1

Schematic of flexible insect flight motor model: (a) Click mechanism, (b) Duffing oscillator model

Grahic Jump Location
Fig. 2

Schematic of lumped vortex model

Grahic Jump Location
Fig. 3

(a) Comparison of peak lift coefficients, estimated using LVM (present method) and from Navier–Stokes simulation ([23]) and (b) comparison of relative phase angles between the lift coefficient and the heaving motion (ϕL), estimated using LVM (present method) and Navier–Stokes results [23]

Grahic Jump Location
Fig. 4

Temporal convergence of coupled system responses at F¯=0.08

Grahic Jump Location
Fig. 5

Displacement time histories and corresponding phase portraits: F¯=0 (a) and (d), F¯=0.02 (b) and (e), F¯=0.08 (c) and (f)

Grahic Jump Location
Fig. 6

Transient chaos at F¯=0.14: (a) displacement time history and (b) phase portrait; period doubling cascade: (c) zoomed period-2 response and (e) period-2 attractor for F¯=0.14, (d) zoomed period-4 response and (f) period-4 attractor for F¯=0.146

Grahic Jump Location
Fig. 7

Transient chaos at F¯=0.16: (a) displacement time history, (b) frequency spectra, (c) zoomed window of chaotic transient, (d) zoomed window of period-1 oscillation, (e) phase portrait and Poincaré section for chaotic transient, and (f) phase portrait and Poincaré section for period-1 oscillation

Grahic Jump Location
Fig. 8

Periodic windows: (a) period-1 oscillation and (b) period-1 attractor at F¯=0.20; (c) period-5 oscillation and (d) period-5 attractor at F¯=0.212

Grahic Jump Location
Fig. 9

Intermittent time history of displacement at F¯=0.232

Grahic Jump Location
Fig. 10

Sustained chaos at F¯=0.24: (a) displacement time history, (b) frequency spectra, (c) zoomed window of chaotic response, and (d) phase portrait

Grahic Jump Location
Fig. 11

Poincaré sections of the response at F¯=0.24: strange attractors

Grahic Jump Location
Fig. 12

Largest Lyapunov exponent of the response at F¯=0.24

Grahic Jump Location
Fig. 13

RP for the period-1 response at F¯=0.08

Grahic Jump Location
Fig. 14

RP for the transient chaos at F¯=0.16

Grahic Jump Location
Fig. 15

RP for the intermittent response at F¯=0.232

Grahic Jump Location
Fig. 16

RP for the chaotic response at F¯=0.24

Grahic Jump Location
Fig. 17

Scalogram of the wavelet coefficients: (a) period-1 response at F¯=0.08, (b) transient chaos at F¯=0.16, (c) intermittent response at F¯=0.232, and (d) chaotic response at F¯=0.24

Grahic Jump Location
Fig. 18

Sustained chaotic behavior of lift coefficient at F¯=0.24: (a) time history of Cl, (b) frequency spectra, (c) zoomed time history, and (d) phase portrait and Poincaré section

Grahic Jump Location
Fig. 19

Transition in the flow pattern: (a) Periodic flow pattern at F¯=0.20 and (b) chaotic flow pattern at F¯=0.24




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In