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Hovering flight helps facilitate feeding, pollination, and courtship. Observed only in smaller flying animals, hover kinematic characteristics are diverse except for the decreasing flapping frequency with the animal size. Although studies have shown that these wing patterns enable distinct unsteady aerodynamic mechanisms, the role of flapping frequency scaling remains a source of disagreement. Here we show that negative allometry of the flapping frequency is required to sustain body attitude during hovering, consistent with experimental data of hovering animals, from fruit flies to hummingbirds, reported in the literature. The derived scaling model reveals that the lift coefficient and reduced frequency remain invariant with mass, enabling leading-edge vortex formation and wake-capture for a wide range of fliers to hover. 

Abstract Flapping flight of animals has captured the interest of researchers due to their impressive flight capabilities across diverse environments including mountains, oceans, forests, and urban areas. Despite the significant progress made in understanding flapping flight, high-altitude flight as showcased by many migrating animals remains underexplored. At high-altitudes, air density is low, and it is challenging to produce lift. Here we demonstrate a first lift-off of a flapping wing robot in a low-density environment through wing size and motion scaling. Force measurements showed that the lift remained high at 0.14 N despite a 66% reduction of air density from the sea-level condition. The flapping amplitude increased from 148 to 233 degrees, while the pitch amplitude remained nearly constant at 38.2 degrees. The combined effect is that the flapping-wing robot benefited from the angle of attack that is characteristic of flying animals. Our results suggest that it is not a simple increase in the flapping frequency, but a coordinated increase in the wing size and reduction in flapping frequency enables the flight in lower density condition. The key mechanism is to preserve the passive rotations due to wing deformation, confirmed by a bioinspired scaling relationship. Our results highlight the feasibility of flight under a low-density, high-altitude environment due to leveraging unsteady aerodynamic mechanisms unique to flapping wings. We anticipate our experimental demonstration to be a starting point for more sophisticated flapping wing models and robots for autonomous multi-altitude sensing. Furthermore, it is a preliminary step towards flapping wing flight in the ultra-low density Martian atmosphere. 

The long-range migration of monarch butterflies, extended over 4000 km, is not well understood. Monarchs experience varying density conditions during migration, ranging as high as 3000 m, where the air density is much lower than at sea level. In this study, we test the hypothesis that the aerodynamic performance of monarchs improves at reduced density conditions by considering the fluid–structure interaction of chordwise flexible wings. A well-validated, fully coupled Navier–Stokes/structural dynamics solver was used to illustrate the interplay between wing motion, aerodynamics, and structural flexibility in forward flight. The wing density and elastic modulus were measured from real monarch wings and prescribed as inputs to the aeroelastic framework. Our results show that sufficient lift is generated to offset the butterfly weight at higher altitudes, aided by the wake-capture mechanism, which is a nonlinear wing–wake interaction mechanism, commonly seen for hovering animals. The mean total power, defined as the sum of the aerodynamic and inertial power, decreased by 36% from the sea level to the condition at 3000 m. Decreasing power with altitude, while maintaining the same equilibrium lift, suggests that the butterflies generate lift more efficiently at higher altitudes. 

The annual migration of monarch butterflies, Danaus plexippus, from their summer breeding grounds in North America to their overwintering sites in Mexico can span over 4000 kilometers. Little is known about the aerodynamic mechanism behind this extended flight. This study is motivated by the hypothesis that their flapping wing flight is enhanced by fluid-structure interactions. The objective of this study to quantify the aeroelastic performance of monarch butterfly wings and apply those values in the creation of an artificial wing with an end goal of creating a biomimetic micro-air vehicle. A micro-CT scan, force-deflection measurements, and a finite element solver on real monarch butterfly wings were used to determine the density and elastic modulus. These structural parameters were then used to create a monarch butterfly inspired artificial wing. A solidification process was used to adhere 3D printed vein structures to a membrane. The performance of the artificial butterfly wing was tested by measuring the lift at flapping frequencies between 6.3 and 14 Hz. Our results show that the elastic modulus of a real wing is 1.8 GPa along the span and 0.20 GPa along the chord, suggesting that the butterfly wing material is highly anisotropic. Real right forewings performed optimally at approximately 10 Hz, the flapping frequency of a live monarch butterfly, with a peak force of 4 mN. The artificial wing performed optimally at approximately 8 Hz with a peak force of 5 mN. 

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.