Experimental studies suggest differences in the distribution of thorax elasticity between insects with synchronous and asynchronous musculature.

Insects have developed diverse flight actuation mechanisms, including synchronous and asynchronous musculature. Indirect actuation, used by insects with both synchronous and asynchronous musculature, transforms thorax exoskeletal deformation into wing rotation. Though thorax deformation is often attributed exclusively to muscle tension, the inertial and aerodynamic forces generated by the flapping wings may also contribute. In this study, a tethered flight experiment was used to simultaneously measure thorax deformation and the inertial/aerodynamic forces acting on the thorax generated by the flapping wing. Compared to insects with synchronous musculature, insects with asynchronous muscle deformed their thorax 60% less relative to their thorax diameter and their wings generated 2.8 times greater forces relative to their body weight. In a second experiment, dorsalventral thorax stiffness was measured across species. Accounting for weight and size, the asynchronous thorax was on average 3.8 times stiffer than the synchronous thorax in the dorsalventral direction. Differences in thorax stiffness and forces acting at the wing hinge led us to hypothesize about differing roles of series and parallel elasticity in the thoraxes of insects with synchronous and asynchronous musculature. Specifically, wing hinge elasticity may contribute more to wing motion in insects with asynchronous musculature than in those with synchronous musculature.

Investigation of chordwise functionally graded flexural rigidity in flapping wings using a two-dimensional pitch-plunge model.

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.

Structural dynamics of real and modelled Solanum stamens: implications for pollen ejection by buzzing bees.

An estimated 10% of flowering plant species conceal their pollen within tube-like anthers that dehisce through small apical pores (poricidal anthers). Bees extract pollen from poricidal anthers through a complex motor routine called floral buzzing, whereby the bee applies vibratory forces to the flower stamen by rapidly contracting its flight muscles. The resulting deformation depends on the stamen's natural frequencies and vibration mode shapes, yet for most poricidal species, these properties have not been sufficiently characterized. We performed experimental modal analysis on Solanum elaeagnifolium stamens to quantify their natural frequencies and vibration modes. Based on morphometric and dynamic measurements, we developed a finite-element model of the stamen to identify how variable material properties, geometry and bee weight could affect its dynamics. In general, stamen natural frequencies fell outside the reported floral buzzing range, and variations in stamen geometry and material properties were unlikely to bring natural frequencies within this range. However, inclusion of bee mass reduced natural frequencies to within the floral buzzing frequency range and gave rise to an axial-bending vibration mode. We hypothesize that floral buzzing bees exploit the large vibration amplification factor of this mode to increase anther deformation, which may facilitate pollen ejection.

Modeling and Analysis of a Simple Flexible Wing-Thorax System in Flapping-Wing Insects.

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.

The flying insect thoracic cuticle is heterogenous in structure and in thickness-dependent modulus gradation.

The thorax is a specialized structure central to insect flight. In the thorax, flight muscles are surrounded by a thin layer of cuticle. The structure, composition, and material properties of this chitinous structure may influence the efficiency of the thorax in flight. However, these properties, as well as their variation throughout the thorax and between insect taxa, are not known. We provide a multi-faceted assessment of thorax cuticle for fliers with asynchronous (honey bee; Apis mellifera) and synchronous (hawkmoth; Manduca sexta) muscles. These muscle types are defined by the relationship between their activation frequency and the insect's wingbeat frequency. We investigated cuticle structure using histology, resilin distribution through confocal laser scanning microscopy, and modulus gradation with nanoindentation. Our results suggest that thorax cuticle properties are highly dependent on anatomical region and species. Modulus gradation, but not mean modulus, differed between the two types of fliers. In some regions, A. mellifera had a positive linear modulus gradient from cuticle interior to exterior of about 2 GPa. In M. sexta, modulus values through cuticle thickness were not well represented by linear fits. We utilized finite element modeling to assess how measured modulus gradients influenced maximum stress in cuticle. Stress was reduced when cuticle with a linear gradient was compressed from the high modulus side. These results support the protective role of the A. mellifera thorax cuticle. Our multi-faceted assessment advances our understanding of thorax cuticle structural and material heterogeneity and the potential benefits of material gradation to flying insects. STATEMENT OF SIGNIFICANCE: The insect thorax is essential for efficient flight but questions remain about the contribution of the exoskeletal cuticle. We investigated the microscale properties of the thorax cuticle, a crucial step to determine its role in flight. Techniques including histology, nanoindentation, and confocal laser scanning microscopy revealed that cuticle properties vary through cuticle thickness, by thorax region, and between species with asynchronous (honey bee; Apis mellifera) and synchronous (hawkmoth; Manduca sexta) muscles. This variation highlights the importance of high resolution cuticle assessment for flying insect lineages and points to factors that may (modulus gradation) and may not (mean modulus) contribute to different flight forms. Understanding material variation in the thorax may inform design of technologies inspired by insects, such as mobile micro robots.

Carpenter bee thorax vibration and force generation inform pollen release mechanisms during floral buzzing.

Approximately 10% of flowering plant species conceal their pollen within tube-like poricidal anthers. Bees extract pollen from poricidal anthers via floral buzzing, a behavior during which they apply cyclic forces by biting the anther and rapidly contracting their flight muscles. The success of pollen extraction during floral buzzing relies on the direction and magnitude of the forces applied by the bees, yet these forces and forcing directions have not been previously quantified. In this work, we developed an experiment to simultaneously measure the directional forces and thorax kinematics produced by carpenter bees (Xylocopa californica) during defensive buzzing, a behavior regulated by similar physiological mechanisms as floral buzzing. We found that the buzzing frequencies averaged about 130 Hz and were highly variable within individuals. Force amplitudes were on average 170 mN, but at times reached nearly 500 mN. These forces were 30-80 times greater than the weight of the bees tested. The two largest forces occurred within a plane formed by the bees' flight muscles. Force amplitudes were moderately correlated with thorax displacement, velocity and acceleration amplitudes but only weakly correlated with buzzing frequency. Linear models developed through this work provide a mechanism to estimate forces produced during non-flight behaviors based on thorax kinematic measurements in carpenter bees. Based on the buzzing frequencies, individual bee's capacity to vary buzz frequency and predominant forcing directions, we hypothesize that carpenter bees leverage vibration amplification to increase the deformation of poricidal anthers, and hence the amount of pollen ejected.

Measuring the frequency response of the honeybee thorax.

Insects with asynchronous flight muscles are believed to flap at the effective fundamental frequency of their thorax-wing system. Flapping in this manner leverages the natural elasticity of the thorax to reduce the energetic requirements of flight. However, to the best of our knowledge, the fundamental frequency of the insect wing-muscle-thorax system has not been measured. Here, we measure the linear frequency response function (FRF) of honeybee Apis mellifera thoraxes about their equilibrium state in order to determine their fundamental frequencies. FRFs relate the input force to output acceleration at the insect tergum and are acquired via a mechanical vibration shaker assembly. When compressed 50 µm, the thorax fundamental frequency averaged across all subjects was about 50% higher than reported wingbeat frequencies. We suspect that the measured fundamental frequencies are higher in the experiment than during flight due to boundary conditions and posthumous muscle stiffening. Next, we compress the thorax between 100-300 µm in 50 µm intervals to assess the sensitivity of the fundamental frequency to geometric modifications. For all specimens considered, the thorax fundamental frequency increased nearly monotonically with respect to level of compression. This implies that the thorax behaves as a nonlinear hardening spring when subject to large displacements, which we confirmed via static force-displacement testing. While there is little evidence that insects utilize this non-linearity during flight, the hardening characteristic may be emulated by small resonant-type flapping wing micro air vehicles to increase flapping frequency bandwidth. Overall, methods established through this work provide a foundation for further dynamical studies on insect thoraxes moving forward.

Reconstructing full-field flapping wing dynamics from sparse measurements.

Flapping insect wings deform during flight. This deformation benefits the insect's aerodynamic force production as well as energetic efficiency. However, it is challenging to measure wing displacement field in flying insects. Many points must be tracked over the wing's surface to resolve its instantaneous shape. To reduce the number of points one is required to track, we propose a physics-based reconstruction method called system equivalent reduction expansion processes to estimate wing deformation and strain from sparse measurements. Measurement locations are determined using a weighted normalized modal displacement method. We experimentally validate the reconstruction technique by flapping a paper wing from 5-9 Hz with 45° and measuring strain at three locations. Two measurements are used for the reconstruction and the third for validation. Strain reconstructions had a maximal error of 30% in amplitude. We extend this methodology to a more realistic insect wing through numerical simulation. We show that wing displacement can be estimated from sparse displacement or strain measurements, and that additional sensors spatially average measurement noise to improve reconstruction accuracy. This research helps overcome some of the challenges of measuring full-field dynamics in flying insects and provides a framework for strain-based sensing in insect-inspired flapping robots.

Wing flexibility reduces the energetic requirements of insect flight.

Flapping insect wings deform under aerodynamic as well as inertial-elastic forces. This deformation is thought to improve power economy and reduce the energetic costs of flight. However, many flapping wing models employ rigid body simplifications or demand excessive computational power, and are consequently unable to identify the influence of flexibility on flight energetics. Here, we derive a reduced-order model capable of estimating the driving torques and corresponding power of flapping, flexible insect wings. We validate this model by actuating a tobacco hornworm hawkmoth Manduca sexta (L.) forewing with a custom single-degree-of-freedom mechanical flapper. Our model predicts measured torques and instantaneous power with reasonable accuracy. Moreover, the flexible wing model predicts experimental trends that rigid body models cannot, which suggests compliance should not be neglected when considering flight dynamics at this scale. Next, we use our model to investigate flight energetics with realistic flapping kinematics. We find that when the natural frequency of the wing is roughly three times that of the flapping frequency, flexibility can reduce energy expenditures by almost 25% compared to a rigid wing if negative work is stored as potential energy and subsequently released to do positive work. The wing itself can store about 30% of the 1200 [Formula: see text]J of total energy required over a wingbeat. Peak potential energy storage occurs immediately before stroke reversal. We estimate that for a moth weighing 1.5-2.5 g, the peak instantaneous power required for flight is 75-125 W kg-1. However, these peak values are likely lower in natural insect flight, where the wing is able to exchange strain energy with the compliant thorax. Our findings highlight the importance of flexibility in flapping wing micro aerial vehicle design and suggest tuned flexibility can greatly improve vehicle efficiency.

Asymmetries in wing inertial and aerodynamic torques contribute to steering in flying insects.

Maneuvering in both natural and artificial miniature flying systems is assumed to be dominated by aerodynamic phenomena. To explore this, we develop a flapping wing model integrating aero and inertial dynamics. The model is applied to an elliptical wing similar to the forewing of the Hawkmoth Manduca sexta and realistic kinematics are prescribed. We scrutinize the stroke deviation phase, as it relates to firing latency in airborne insect steering muscles which has been correlated to various aerial maneuvers. We show that the average resultant force production acting on the body largely arises from wing pitch and roll and is insensitive to the phase and amplitude of stroke deviation. Inclusion of stroke deviation can generate significant averaged aerodynamic torques at steady-state and adjustment of its phase can facilitate body attitude control. Moreover, averaged wing angular momentum varies with stroke deviation phase, implying a non-zero impulse during a time-dependent phase shift. Simulations show wing inertial and aerodynamic impulses are of similar magnitude during short transients whereas aerodynamic impulses dominate during longer transients. Additionally, inertial effects become less significant for smaller flying insects. Body yaw rates arising from these impulses are consistent with biologically measured values. Thus, we conclude (1) modest changes in stroke deviation can significantly affect steering and (2) both aerodynamic and inertial torques are critical to maneuverability, the latter of which has not widely been considered. Therefore, the addition of a control actuator modulating stroke deviation may decouple lift/thrust production from steering mechanisms in flapping wing micro aerial vehicles and increase vehicle dexterity through inertial trajectory shaping.