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Flapping insect wings experience appreciable deformation due to aerodynamic and inertial forces. This deformation is believed to benefit the insect’s aerodynamic force production as well as energetic efficiency. However, the fluid-structure interaction (FSI) models used to estimate wing deformations are often computationally demanding and are therefore challenged by parametric studies. Here, we develop a simple FSI model of a flapping wing idealized as a two-dimensional pitching-plunging airfoil. Using the Lagrangian formulation, we derive the reduced-order structural framework governing wing’s elastic deformation. We consider two fluid models: quasi-steady Deformable Blade Element Theory (DBET) and Unsteady Vortex Lattice Method (UVLM). DBET is computationally economical but does not provide insight into the flow structure surrounding the wing, whereas UVLM approximates flows but requires more time to solve. For simple flapping kinematics, DBET and UVLM produce similar estimates of the aerodynamic force normal to the surface of a rigid wing. More importantly, when the wing is permitted to deform, DBET and UVLM agree well in predicting wingtip deflection and aerodynamic normal force. The most notable difference between the model predictions is a roughly 20° phase difference in normal force. DBET estimates wing deformation and force production approximately 15 times faster than UVLM for the parameters considered, and both models solve in under a minute when considering 15 flapping periods. Moving forward, we will benchmark both low-order models with respect to high fidelity computational fluid dynamics coupled to finite element analysis, and assess the agreement between DBET and UVLM over a broader range of flapping kinematics.

 

Flapping, flexible wings deform under both aerodynamic and inertial loads. However, the fluid-structure interaction (FSI) governing flapping wing dynamics is not well understood. Conventional FSI models require excessive computational resources and are not conducive to parameter studies that consider variable wing kinematics or geometry. Here, we present a simple two-way coupled FSI model for a wing subjected to single-degree-of-freedom (SDOF) rotation. The model is reduced-order and can be solved several orders of magnitude faster than direct computational methods. We construct a SDOF rotation stage and measure basal strain of a flapping wing in-air and in-vacuum to study our model experimentally. Overall, agreement between theory and experiment is excellent. In-vacuum, the wing has a large 3ω response when flapping at approximately 1/3 its natural frequency. This response is attenuated substantially when flapping in-air as a result of aerodynamic damping. These results highlight the importance of two-way coupling between the fluid and structure, since one-way coupled approaches cannot describe such phenomena. Moving forward, our model enables advanced studies of biological flight and facilitates bio-inspired design of flapping wing technologies.

 

Flapping, flexible insect wings deform under inertial and fluid loading. Deformation influences aerodynamic force generation and sensorimotor control, and is thus important to insect flight mechanics. Conventional flapping wing fluid–structure interaction models provide detailed information about wing deformation and the surrounding flow structure, but are impractical in parameter studies due to their considerable computational demands. Here, we develop two quasi three-dimensional reduced-order models (ROMs) capable of describing the propulsive forces/moments and deformation profiles of flexible wings. The first is based on deformable blade element theory (DBET) and the second is based on the unsteady vortex lattice method (UVLM). Both rely on a modal-truncation based structural solver. We apply each model to estimate the aeromechanics of a thin, flapping flat plate with a rigid leading edge, and compare ROM findings to those produced by a coupled fluid dynamics/finite element computational solver. The ROMs predict wing deformation with good accuracy even for relatively large deformations of 25% of the chord length. Aerodynamic loading normal to the wing's rotation plane is well captured by the ROMs, though model errors are larger for in-plane loading. We then perform a parameter sweep to understand how wing flexibility and mass affect peak deflection, mean lift and average power. All models indicate that flexible wings produce less lift but require lower average power to flap. Importantly, these studies highlight the computational efficiency of the ROMs—compared to the convention modeling approach, the UVLM and DBET ROMs solve 4 and 6 orders of magnitude faster, respectively.

 

Small-scale flapping-wing micro air vehicles (FWMAVs) are an emerging robotic technology with many applications in areas including infrastructure monitoring and remote sensing. However, challenges such as inefficient energetics and decreased payload capacity preclude the useful implementation of FWMAVs. Insects serve as inspiration to FWMAV design owing to their energy efficiency, maneuverability, and capacity to hover. Still, the biomechanics of insects remain challenging to model, thereby limiting the translational design insights we can gather from their flight. In particular, it is not well-understood how wing flexibility impacts the energy requirements of flapping flight. In this work, we developed a simple model of an insect drive train consisting of a compliant thorax coupled to a flexible wing flapping with single-degree-of-freedom rotation in a fluid environment. We applied this model to quantify the energy required to actuate a flapping wing system with parameters based off a hawkmoth Manduca sexta. Despite its simplifications, the model predicts thorax displacement, wingtip deflection and peak aerodynamic force in proximity to what has been measured experimentally in flying moths. We found a flapping system with flexible wings requires 20% less energy than a flapping system with rigid wings while maintaining similar aerodynamic performance. Passive wing deformation increases the effective angle of rotation of the flexible wing, thereby reducing the maximum rotation angle at the base of the wing. We investigated the sensitivity of these results to parameter deviations and found that the energetic savings conferred by the flexible wing are robust over a wide range of parameters. 

Abstract Flapping insect wings collide with vegetation and other obstacles during flight. Repeated collisions may irreversibly damage the insect wing, thereby compromising the insect's ability to fly. Further, reaction torques caused by the collision may destabilize the insect and hinder its ability to maneuver. To mitigate the adverse effects of impact, some insect wings are equipped with a flexible joint called a “costal break.” The costal break buckles once it exceeds a critical angle, which is believed to improve flight stability and prevent irreversible wing damage. However, to our knowledge, there are no models to predict the dynamics of the costal break. Through this research, we develop a simple model of an insect wing with a costal break. The wing was modeled as two beams interconnected by a torsional spring, where the stiffness of the torsional spring instantaneously decreases once it has exceeded a critical angle. We conducted a series of static tests to approximate model parameters. Then, we used numerical simulation to estimate the reaction moments, angular impulse, and peak stresses experienced by the wing during a collision. When evaluated over the duration of an external load, we found that buckling could reduce reaction moments and angular impulse up to 82% and 99%, respectively, compared to a homogeneous wing. This suggests the costal break can enhance flight stability. On the other hand, buckling maximally increased peak stresses two times compared to a homogeneous wing, indicating the costal break does not reduce likelihood of damage under the simplified loading considered. 

Abstract Insect wings are heterogeneous structures, with flexural rigidity varying one to two orders of magnitude over the wing surface. This heterogeneity influences the deformation the flapping wing experiences during flight. However, it is not well understood how this flexural rigidity gradient affects wing performance. Here, we develop a simplified 2D model of a flapping wing as a pitching, plunging airfoil using the assumed mode method and unsteady vortex lattice method to model the structural and fluid dynamics, respectively. We conduct parameter studies to explore how variable flexural rigidity affects mean lift production, power consumption and the forces required to flap the wing. We find that there is an optimal flexural rigidity distribution that maximizes lift production; this distribution generally corresponds to a 3:1 ratio between the wing’s flapping and natural frequencies, though the ratio is sensitive to flapping kinematics. For hovering flight, the optimized flexible wing produces 20% more lift and requires 15% less power compared to a rigid wing but needs 20% higher forces to flap. Even when flapping kinematics deviate from those observed during hover, the flexible wing outperforms the rigid wing in terms of aerodynamic force generation and power across a wide range of flexural rigidity gradients. Peak force requirements and power consumption are inversely proportional with respect to flexural rigidity gradient, which may present a trade-off between insect muscle size and energy storage requirements. The model developed in this work can be used to efficiently investigate other spatially variant morphological or material wing features moving forward. 

Flapping wing deformation influences the aerodynamics of insect flight. This deformation is dictated by the dynamical properties of the insect wing, particularly its vibration spectra and mode shapes. However, researchers have not yet developed artificial insect wings with vibration spectra and mode shapes that are identical to their biological counterparts. The goal of the present work is to develop artificial insect wings that are both isospectral and isomodal with respect to real insect wings. To do so, we characterized hawkmoth Manduca sexta wings using experimental modal analyses. From these results, we created artificial wings using additive manufacturing and heat molding. Between artificial and real wings, the first two natural frequencies differ by 7% and 16% respectively, with differences of 16% and 131% in gains evaluated at those natural frequencies. Vibration modes are similar as well. This work provides a foundation for more advanced wing design moving forward. 

Abstract Flapping wings deform under both aerodynamic and inertial forces. However, many flapping wing fluid–structure interaction (FSI) models require significant computational resources which limit their effectiveness for high-dimensional parametric studies. Here, we present a simple bilaterally coupled FSI model for a wing subject to single-degree-of-freedom (SDOF) flapping. The model is reduced-order and can be solved several orders of magnitude faster than direct computational methods. To verify the model experimentally, we construct a SDOF rotation stage and measure basal strain of a flapping wing in-air and in-vacuum. Overall, the derived model estimates wing strain with good accuracy. In-vacuum, the wing has a large 3ω response when flapping at approximately one-third of its natural frequency due to a superharmonic resonance, where the superharmonic occurs due to the interaction of inertial forces and time-varying centrifugal softening. In-air, this 3ω response is attenuated significantly as a result of aerodynamic damping, whereas the primary ω response is increased due to aerodynamic loading. These results highlight the importance of (1) bilateral coupling between the fluid and structure, since unilaterally coupled approaches do not adequately describe deformation-induced aerodynamic damping and (2) time-varying stiffness, which generates superharmonics of the flapping frequency in the wing’s dynamic response. The simple SDOF model and experimental study presented in this work demonstrate the potential for a reduced-order FSI model that considers both bilateral fluid–structure coupling and realistic multi-degrees-of-freedom flapping kinematics moving forward.