Abstract
Natural flyers have flexible wings, which deform significantly under the combined inertial and aerodynamic forces. In this study, we focus on the role of chord wise flexibility in 2D pitch and plunge motions. We derive the exact nonlinear 2D equations of motion for a flexible flapping wing with flying support. In achieving the closed-form equations, we use the exact strain field concerning considerable elastic deformations. After numerically solving the novel equations, we validate them in simulations with highly deformable wings. By coupling the derived equations of motion with fluid flow, we study the aerodynamic performance of the geometrically nonlinear flexible flapping wing. Through numerical simulations, we see that a flexible wing generates much lower drag with increasing flexibility. While at angles of attack \(>\) \(50^{\circ }\), more lift is generated by flexible wings.
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References
Dickinson, M.H.: Bionics: biological insight into mechanical design. Proc. Nat. Acad. Sci. USA 96(25), 14208–14209 (1999)
Hedrick, T.L., Cheng, B., Deng, X.: Wingbeat time and the scaling of passive rotational damping in flapping flight. Science 234, 252–255 (2009)
Lasek, M., Pietrucha, J., Sibilski, K., Zlocka, M.: A study of flight dynamics and automatic control of an animalopter. ICAS 2002 Congress 553, 1–8 (2002)
Liang, B., Sun, M.: Nonlinear flight dynamics and stability of hovering model insects. J. R. Soc. Interface 10(85), 20130269 (2013)
Taha, H.E., Hajj, M.R., Nayfeh, A.H.: Flight dynamics and control of flapping-wing mavs: a review. Nonlinear Dyn. 70(2), 907–939 (2012)
Ma, K.Y., Chirarattananon, P., Fuller, S.B., Wood, R.J.: Controlled flight of a biologically inspired, insect-scale robot. Science 340(6132), 603–607 (2013)
Daniel, T.L., Combes, S.A.: Flexible wings and fins: bending by inertial or fluid-dynamic forces. Integr. Comp. Biol. 42, 1044–1049 (2002)
Lin, C.S., Hwu, C., Young, W.B.: The thrust and lift of an ornithopters membrane wings with simple flapping motion. Aerosp. Sci. Technol. 10, 111–119 (2006)
Usherwood, J.R., Hedrick, T.L., McGowan, C.P., Biewener, A.A.: Dynamic pressure maps for wings and tails of pigeons in slow, flapping flight, and their energetic implications. J. Exp. Biol. 208, 355–369 (2005)
Zhu, J., Zhou, C., Wang, C., Jiang, L.: Effect of flexibility on flapping wing characteristics under forward flight. Fluid Dyn. Res. 46(5), 055515 (2014)
Dickinson, M.H., Lehmann, F.O., Sane, S.P.: Wing rotation and the aerodynamic basis of insect flight. Science 284, 1954–1960 (1999)
Taha, H.E., Tahmasian, S., Woolsey, C.A., Nayfeh, A.H., Hajj, M.R.: The need for higher-order averaging in the stability analysis of hovering, flapping-wing flight. Bioinspir. Biomim. 10(1), 016,002 (2015)
Wu, J.H., Zhang, Y.L., Sun, M.: Hovering of model insects: simulation by coupling equations of motion with navier-stokes equations. J. Exp. Biol. 212(20), 3313–3329 (2009)
Ifju, P.G., Jenkins, D.A., Ettinger, S., Lian, Y., Shyy, W.: Flexible-wing-based micro air vehicles. AIAA (2002-0705), 1–13 (2002)
Raney, D.L., Waszak, M.R.: Biologically inspired micro-flight research. SAE Trans. 112(1), 598–610 (2003)
Tamai, M., Murphy, J.T., Hu, H.: An experimental study of flexible membrane airfoils at low reynolds numbers. AIAA 46th Aerospace Sciences Meeting and Exhibit AIAA-2008-0580, 1–12 (2008)
Heathcote, S., Wang, Z., Gursul, I.: Effect of spanwise flexibility on flapping wing propulsion. J. Fluids Struct. 24(2), 183–199 (2008)
Tian, F.B., Luo, H., Song, J., Lu, X.Y.: Force production and asymmetric deformation of a flexible flapping wing in forward flight. J. Fluids Struct. 36, 149–161 (2013)
Kang, C., Shyy, W.: Analytical model for instantaneous lift and shape deformation of an insect-scale flapping wing in hover. J. R. Soc. Interface 11(101), 20140,933 (2014)
Cho, H., Kwak, J., Shin, S.: Computational analysis for flapping wing by coupling the geometrically exact beam and preconditioned Navier-stokes solution. The 55th AIAA/ASME/ASCE/AHS/SC Structures, Structural Dynamics, and Materials Conference, National Harbor, 13–17 Jan 2014
Lacarbonara, W., Yabuno, H.: Refined models of elastic beams undergoing large in-plane motions: theory and experiment. Int. J. Solids Struct. 43(17), 5066–5084 (2006)
Nayfeh, A.H., Pai, P.F.: Linear and Nonlinear Structural Mechanics. Wiley, Hoboken, New Jersey (2004)
Abedinnasab, M., Zohoor, H., Yoon, Y.: Exact formulations of non-linear planar and spatial Euler–Bernoulli beams with finite strains. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 226(5), 1225–1236 (2012)
Liao, S.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. CRC Press, Boca Raton (2004)
Abedinnasab, M.H., Hussein, M.I.: Wave dispersion under finite deformation. Wave motion 50(3), 374–388 (2013)
COMSOL Multiphysics 4.2.: COMSOL Inc., Palo Alto, CA (2011)
Neef, M., Hummel, D.: Euler solutions for a finite-span flapping wing. In: Fixed, flapping and rotary vehicles at very low Reynolds Numbers, Proceedings of the Conference, Notre Dame, IN, pp. 75–99 (2000)
Donea, J., Huerta, A., Ponthot, J.-P., Rodríguez-Ferran, A.: Arbitrary Lagrangian–Eulerian methods. The Encyclopedia of Computational Mechanics. Wiley, New York (2004)
Schenk, O., Gärtner, K.: Solving unsymmetric sparse systems of linear equations with pardiso. Future Gener. Comput. Syst. 20(3), 475–487 (2004)
Taira, K., Colonius, T.: Three-dimensional flows around low-aspect-ratio flat-plate wings at low reynolds numbers. J. Fluid Mech. 623, 187–207 (2009)
Madangopal, R., Khan, Z.A., Agrawal, S.K.: Biologically inspired design of small flapping wing air vehicles using four-bar mechanisms and quasi-steady aerodynamics. ASME J. Mech. Des. 127, 809–816 (2005)
Bungartz, H.J., Mehl, M., Schäfer, M.: Fluid Structure Interaction II: Modelling, Simulation, Optimization, vol. 73. Springer, Berlin (2010)
Zhao, L., Huang, Q., Deng, X., Sane, S.P.: Aerodynamic effects of flexibility in flapping wings. J. R. Soc. Interface 7, 485–497 (2009)
Mazaheri, K., Ebrahimi, A.: Experimental investigation of the effect of chordwise flexibility on the aerodynamics of flapping wings in hovering flight. J. Fluids Struct. 26, 544–558 (2010)
Hamamoto, M., Ohta, Y., Hisada, T.: A fundamental study of wing actuation for a 6-in-wingspan flapping microaerial vehicle. IEEE Trans. Robot. 26(2), 244–255 (2010)
Shyy, W., et al.: Computational aerodynamics of low reynolds number plunging, pitching and flexible wings for MAV applications. Acta Mech. Sin. 24, 351–373 (2008)
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Abedinnasab, M.H., Zohoor, H. & Yoon, YJ. Nonlinear analysis of 2D flexible flapping wings. Nonlinear Dyn 81, 299–310 (2015). https://doi.org/10.1007/s11071-015-1992-6
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DOI: https://doi.org/10.1007/s11071-015-1992-6