Analytical model for instantaneous lift and shape deformation of an insect-scale flapping wing in hover

Abstract

In the analysis of flexible flapping wings of insects, the aerodynamic outcome depends on the combined structural dynamics and unsteady fluid physics. Because the wing shape and hence the resulting effective angle of attack are a priori unknown, predicting aerodynamic performance is challenging. Here, we show that a coupled aerodynamics/structural dynamics model can be established for hovering, based on a linear beam equation with the Morison equation to account for both added mass and aerodynamic damping effects. Lift strongly depends on the instantaneous angle of attack, resulting from passive pitch associated with wing deformation. We show that both instantaneous wing deformation and lift can be predicted in a much simplified framework. Moreover, our analysis suggests that resulting wing kinematics can be explained by the interplay between acceleration-related and aerodynamic damping forces. Interestingly, while both forces combine to create a high angle of attack resulting in high lift around the midstroke, they offset each other for phase control at the end of the stroke.

Aerodynamics, sensing and control of insect-scale flapping-wing flight

“…Specifically, the instantaneous lift follows from the quasi-steady model [26,74] based on the corresponding AoA due to wing deformations. Figure 11 depicts the outcome of the quasi-steady model for flexible wings with the angles of attack due to passive pitch α and lift coefficients C L as a function of time [73]. Both α and C L agree well with high-fidelity numerical results [74].…”

“…The dynamic flight stability is an inherent property of a flapping insect [137] with stabilization, manoeuvre and steady-state control requiring the instantaneous information about the unsteady aerodynamics and time-varying body and wing motions [137]. Under the assumption of small wing deformations, the combined fluid-structure coupling simplifies to a dynamic balance between chordwise wing deformation and drag [73]. The unsteady aerodynamic force can be modelled with the Morison equation [138] that accounts for both added mass and aerodynamic damping.…”

“…An adjusted quasi-steady model by correcting the surrounding flow parameters including the effective AoA and associated flow structures such as jet flows, can perform much better. By assuming that the instantaneous aerodynamic forces depend on the instantaneous velocity and acceleration of the wing motion, these quasi-steady models provide a closed form formulation for the delayed stall, rotational circulation and added mass [26,27,70,71] including passively rotating wings [72] and chordwise flexible aerofoils [73].…”

“…These coefficients are empirically determined from various experiments [26,27,73,76]. Such a need for empirically fitted coefficients can be partially mitigated by considering an unsteady model that captures the nonlinear effect of the LEV on the lift [77] or a hybrid model of high-fidelity Navier-Stokes equation solutions and low-order quasi-steady models [78].…”

“…the wing undergoes unsteady motions [26] and aerodynamic [75] or aeroelastic [73] damping coefficients. These coefficients are empirically determined from various experiments [26,27,73,76].…”