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 area prioriunknown, 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.

We mimic a flapping wing through a fluid–structure interaction (FSI) framework based upon a generalized lumped-torsional flexibility model. The developed fluid and structural solvers together determine the aerodynamic forces, wing deformation and self-propelled motion. A phenomenological solution to the linear single-spring structural dynamics equation is established to help offer insight and validate the computations under the limit of small deformation. The cruising velocity and power requirements are evaluated by varying the flapping Reynolds number ( $20\leqslant Re_{f}\leqslant 100$ ), stiffness (represented by frequency ratio, $1\lesssim \unicode[STIX]{x1D714}^{\ast }\leqslant 10$ ) and the ratio of aerodynamic to structural inertia forces (represented by a dimensionless parameter $\unicode[STIX]{x1D713}$ ( $0.1\leqslant \unicode[STIX]{x1D713}\leqslant 3$ )). For structural inertia dominated flows ( $\unicode[STIX]{x1D713}\ll 1$ ), pitching and plunging are shown to always remain in phase ( $\unicode[STIX]{x1D719}\approx 0$ ) with the maximum wing deformation occurring at the end of the stroke. When aerodynamics dominates ( $\unicode[STIX]{x1D713}>1$ ), a large phase difference is induced ( $\unicode[STIX]{x1D719}\approx \unicode[STIX]{x03C0}/2$ ) and the maximum deformation occurs at mid-stroke. Lattice Boltzmann simulations show that there is an optimal $\unicode[STIX]{x1D714}^{\ast }$ at which cruising velocity is maximized and the location of optimum shifts away from unit frequency ratio ( $\unicode[STIX]{x1D714}^{\ast }=1$ ) as $\unicode[STIX]{x1D713}$ increases. Furthermore, aerodynamics administered deformations exhibit better performance than those governed by structural inertia, quantified in terms of distance travelled per unit work input. Closer examination reveals that although maximum thrust transpires at unit frequency ratio, it is not transformed into the highest cruising velocity. Rather, the maximum velocity occurs at the condition when the relative tip displacement ${\approx}\,0.3$ .

Achieving atmospheric flight on Mars is challenging due to the low density of the Martian atmosphere. Aerodynamic forces are proportional to the atmospheric density, which limits the use of conventional aircraft designs on Mars. Here, we show using numerical simulations that a flapping wing robot can fly on Mars via bioinspired dynamic scaling. Trimmed, hovering flight is possible in a simulated Martian environment when dynamic similarity with insects on earth is achieved by preserving the relevant dimensionless parameters while scaling up the wings three to four times its normal size. The analysis is performed using a well-validated 2D Navier-Stokes equation solver, coupled to a 3D flight dynamics model to simulate free flight. The majority of power required is due to the inertia of the wing because of the ultra-low density. The inertial flap power can be substantially reduced through the use of a torsional spring. The minimum total power consumption is 188 W kg-1 when the torsional spring is driven at its natural frequency.

Wing-wake interaction is a characteristic nonlinear flow feature that can enhance unsteady lift in flapping flight. However, the effects of wing-wake interaction on the flight dynamics of hover are inadequately understood. We use a well-validated 2D Navier-Stokes equation solver and a quasi-steady model to investigate the role of wing-wake interaction on the hover stability of a fruit fly scale flapping flyer. The Navier-Stokes equations capture wing-wake interaction, whereas the quasi-steady models do not. Both aerodynamic models are tightly coupled to a flight dynamic model, which includes the effects of wing mass. The flapping amplitude, stroke plane angle, and flapping offset angle are adjusted in free flight for various wing rotations to achieve hover equilibrium. We present stability results for 152 simulations which consider different kinematics involving the pitch amplitude and pitch axis as well as the duration and timing of pitch rotation. The stability of all studied motions was qualitatively similar, with an unstable oscillatory mode present in each case. Wing-wake interaction has a destabilizing effect on the longitudinal stability, which cannot be predicted by a quasi-steady model. Wing-wake interaction increases the tendency of the flapping flyer to pitch up in the presence of a horizontal velocity perturbation, which further destabilizes the unstable oscillatory mode of hovering flight dynamics.

Various tools have been developed to model the aerodynamics of flapping wings. In particular, quasi-steady models, which are considerably faster and easier to solve than the Navier–Stokes equations, are often utilized in the study of flight dynamics of flapping wing flyers. However, the accuracy of the quasi-steady models has not been properly documented. The objective of this study is to assess the accuracy of a quasi-steady model by comparing the resulting aerodynamic forces against three-dimensional (3D) Navier–Stokes solutions. The same wing motion is prescribed at a fruit fly scale. The pitching amplitude, axis, and duration are varied. Comparison of the aerodynamic force coefficients suggests that the quasi-steady model shows significant discrepancies under extreme pitching motions, i.e., the pitching motion is large, quick, and occurs about the leading or trailing edge. The differences are as large as 1.7 in the cycle-averaged lift coefficient. The quasi-steady model performs well when the kinematics are mild, i.e., the pitching motion is small, long, and occurs near the mid-chord with a small difference in the lift coefficient of 0.01. Our analysis suggests that the main source for the error is the inaccuracy of the rotational lift term and the inability to model the wing-wake interaction in the quasi-steady model.

Insect wings are flexible, and the dynamically deforming wing shape influences the resulting aerodynamics and power consumption. However, the influence of wing flexibility on the flight dynamics of insects is unknown. Most stability studies in the literature consider rigid wings and conclude that the hover equilibrium condition is unstable. The rigid wings possess an unstable oscillatory mode mainly due to their pitch sensitivity to horizontal velocity perturbations. Here, we show that a flapping wing flyer with flexible wings exhibits stable hover equilibria. The free-flight insect flight dynamics are simulated at the fruit fly scale in the longitudinal plane. The chordwise wing flexibility is modelled as a linear beam. The two-dimensional Navier–Stokes equations are solved in a tight fluid–structure integration scheme. For a range of wing flexibilities similar to live insects, all eigenvalues of the system matrix about the hover equilibrium have negative real parts. Flexible wings appear to stabilize the unstable mode by passively deforming their wing shape in the presence of perturbations, generating significantly more horizontal velocity damping and pitch rate damping. These results suggest that insects may passively stabilize their hover flight via wing flexibility, which can inform designs of synthetic flapping wing robots.