Skip to main content Accessibility help

Bistability in the rotational motion of rigid and flexible flyers

  • Yangyang Huang (a1), Leif Ristroph (a2), Mitul Luhar (a1) and Eva Kanso (a1)


We explore the rotational stability of hovering flight in an idealized two-dimensional model. Our model is motivated by an experimental pyramid-shaped object (Weathers et al., J. Fluid Mech, vol. 650, 2010, pp. 415–425; Liu et al., Phys. Rev. Lett., vol. 108, 2012, 068103) and a computational $\wedge$ -shaped analogue (Huang et al., Phys. Fluids, vol. 27 (6), 2015, 061706; Huang et al., J. Fluid Mech., vol. 804, 2016, pp. 531–549) hovering passively in oscillating airflows; both systems have been shown to maintain rotational balance during free flight. Here, we attach the $\wedge$ -shaped flyer at its apex in oscillating flow, allowing it to rotate freely akin to a pendulum. We use computational vortex sheet methods and we develop a quasi-steady point-force model to analyse the rotational dynamics of the flyer. We find that the flyer exhibits stable concave-down ( $\wedge$ ) and concave-up ( $\vee$ ) behaviour. Importantly, the down and up configurations are bistable and co-exist for a range of background flow properties. We explain the aerodynamic origin of this bistability and compare it to the inertia-induced stability of an inverted pendulum oscillating at its base. We then allow the flyer to flap passively by introducing a rotational spring at its apex. For stiff springs, flexibility diminishes upward stability but as stiffness decreases, a new transition to upward stability is induced by flapping. We conclude by commenting on the implications of these findings for biological and man-made aircraft.


Corresponding author

Email address for correspondence:


Hide All
Alben, S. 2009 Simulating the dynamics of flexible bodies and vortex sheets. J. Comput. Phys. 228 (7), 25872603.
Alben, S. 2010 Flexible sheets falling in an inviscid fluid. Phys. Fluids 22 (6), 061901.
Butikov, E. I. 2001 On the dynamic stabilization of an inverted pendulum. Am. J. Phys. 69 (7), 755768.
Chen, Y., Wang, H., Helbling, E. F., Jafferis, N. T., Zufferey, R., Ong, A., Ma, K., Gravish, N., Chirarattananon, P., Kovac, M. & Wood, R. J. 2017 A biologically inspired, flapping-wing, hybrid aerial-aquatic microrobot. Sci. Robot. 2 (11), 111.
Childress, S., Vandenberghe, N. & Zhang, J. 2006 Hovering of a passive body in an oscillating airflow. Phys. Fluids 18 (11), 117103.
Dickinson, M. H., Lehmann, F.-O. & Sane, S. P. 1999 Wing rotation and the aerodynamic basis of insect flight. Science 284 (5422), 19541960.
Ellington, C. P. 1985 Power and efficiency of insect flight muscle. J. Expl Biol. 115 (1), 293304.
Ellington, C. P., van den Berg, C., Willmott, A. P. & Thomas, A. L. R. 1996 Leading-edge vortices in insect flight. Nature 384 (6610), 626630.
Fang, F., Ho, K. L., Ristroph, L. & Shelley, M. J. 2017 A computational model of the flight dynamics and aerodynamics of a jellyfish-like flying machine. J. Fluid Mech. 819, 621655.
Feldman, A. G. & Levin, M. F. 2009 Progress in Motor Control, vol. 629. Springer.
Fry, S. N., Sayaman, R. & Dickinson, M. H. 2003 The aerodynamics of free-flight maneuvers in drosophila. Science 300 (5618), 495498.
Graule, M. A., Chirarattananon, P., Fuller, S. B., Jafferis, N. T., Ma, K. Y., Spenko, M., Kornbluh, R. & Wood, R. J. 2016 Perching and takeoff of a robotic insect on overhangs using switchable electrostatic adhesion. Science 352 (6288), 978982.
Huang, Y. & Kanso, E. 2015 Periodic and chaotic flapping of insectile wings. Eur. Phys. J. Special Topics 224 (17–18), 31753183.
Huang, Y., Nitsche, M. & Kanso, E. 2015 Stability versus maneuverability in hovering flight. Phys. Fluids 27 (6), 061706.
Huang, Y., Nitsche, M. & Kanso, E. 2016 Hovering in oscillatory flows. J. Fluid Mech. 804, 531549.
Jones, M. A. 2003 The separated flow of an inviscid fluid around a moving flat plate. J. Fluid Mech. 496, 405441.
Jones, M. A. & Shelley, M. J. 2005 Falling cards. J. Fluid Mech. 540, 393425.
Keulegan, H. & Carpenter, L. H. 1958 Forces on cylinders and plates in an oscillating fluid. J. Res. Natl Bur. Stand. 60 (1), 423440.
Krasny, R. 1986 Desingularization of periodic vortex sheet roll-up. J. Comput. Phys. 65 (2), 292313.
Liu, B., Ristroph, L., Weathers, A., Childress, S. & Zhang, J. 2012 Intrinsic stability of a body hovering in an oscillating airflow. Phys. Rev. Lett. 108, 068103.
Ma, K. Y., Chirarattananon, P., Fuller, S. B. & Wood, R. J. 2013 Controlled flight of a biologically inspired, insect-scale robot. Science 340 (6132), 603607.
Nitsche, M. & Krasny, R. 1994 A numerical study of vortex ring formation at the edge of a circular tube. J. Fluid Mech. 276, 139161.
Pringle, J. W. S. 2003 Insect Flight, vol. 9. Cambridge University Press.
Ristroph, L., Bergou, A. J., Ristroph, G., Coumes, K., Berman, G. J., Guckenheimer, J., Wang, Z. J. & Cohen, I. 2010 Discovering the flight autostabilizer of fruit flies by inducing aerial stumbles. Proc. Natl Acad. Sci. USA 107 (11), 48204824.
Ristroph, L. & Childress, S. 2014 Stable hovering of a jellyfish-like flying machine. J. R. Soc. Interface 11 (92), 20130992.
Sane, S. P. 2003 The aerodynamics of insect flight. J. Expl Biol. 206 (23), 41914208.
Sheng, J. X., Ysasi, A., Kolomenskiy, D., Kanso, E., Nitsche, M. & Schneider, K. 2012 Simulating vortex wakes of flapping plates. In Natural Locomotion in Fluids and on Surfaces (ed. Childress, S., Hosoi, A., Schultz, W. W. & Wang, J.), pp. 255262. Springer.
Shukla, R. K. & Eldredge, J. D. 2007 An inviscid model for vortex shedding from a deforming body. Theor. Comput. Fluid Dyn. 21 (5), 343368.
Spedding, G. R., Rosén, M. & Hedenström, A. 2003 A family of vortex wakes generated by a thrush nightingale in free flight in a wind tunnel over its entire natural range of flight speeds. J. Expl Biol. 206 (14), 23132344.
Sun, M. 2014 Insect flight dynamics: stability and control. Rev. Mod. Phys. 86, 615646.
Taylor, G. K. & Krapp, H. G. 2007 Sensory systems and flight stability: what do insects measure and why? Adv. Insect Physiol. 34, 231316.
Thomas, A. L. R., Taylor, G. K., Srygley, R. B., Nudds, R. L. & Bomphrey, R. J. 2004 Dragonfly flight: free-flight and tethered flow visualizations reveal a diverse array of unsteady lift-generating mechanisms, controlled primarily via angle of attack. J. Expl Biol. 207 (24), 42994323.
Vogel, S. 2009 Glimpses of Creatures in Their Physical Worlds. Princeton University Press.
Wang, Z. J. 2005 Dissecting insect flight. Annu. Rev. Fluid Mech. 37 (1), 183210.
Wang, Z. J., Birch, J. M. & Dickinson, M. H. 2004 Unsteady forces and flows in low Reynolds number hovering flight: two-dimensional computations vs robotic wing experiments. J. Expl Biol. 207 (3), 449460.
Warrick, D. R., Tobalske, B. W. & Powers, D. R. 2005 Aerodynamics of the hovering hummingbird. Nature 435 (7045), 10941097.
Weathers, A., Folie, B., Liu, B., Childress, S. & Zhang, J. 2010 Hovering of a rigid pyramid in an oscillatory airflow. J. Fluid Mech. 650, 415425.
Wright, O. & Wright, W.1906 Flying-machine. US Patent 821 393.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed