Distinct forms of resonant optimality within insect indirect flight motors

Abstract

Insect flight motors are extraordinary natural structures that operate efficiently at high frequencies. Structural resonance is thought to play a role in ensuring efficient motor operation, but the details of this role are elusive. While the efficiency benefits associated with resonance may be significant, a range of counterintuitive behaviours are observed. In particular, the relationship between insect wingbeat frequencies and thoracic natural frequencies is uncertain, with insects showing wingbeat frequency modulation over both short and long time scales. Here, we offer new explanations for this modulation. We show how, in linear and nonlinear models of an indirect flight motor, resonance is not a unitary state at a single frequency, but a complex cluster of distinct and mutually exclusive states, each representing a different form of resonant optimality. Additionally, by characterizing the relationship between resonance and the state of negative work absorption within the motor, we demonstrate how near-perfect resonant energetic optimality can be maintained over significant wingbeat frequency ranges. Our analysis leads to a new conceptual model of flight motor operation: one in which insects are not energetically restricted to a precise wingbeat frequency, but instead are robust to changes in thoracic and environmental properties—an illustration of the extraordinary robustness of these natural motors.

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

“…Instead, the greatest mean aerodynamic force occurs at a flap frequency that corresponds to the highest thorax velocity. The idea that maximum thorax displacement and velocity occur at different peak frequencies is consistent with the results of [ 19 ], which show that multiple peak frequencies exist in a linear flapping-wing system depending on the specific transfer function (e.g., force-to-displacement and force-to-velocity) considered. Similar to thorax displacement, the flap frequency corresponding to maximum energy input tends to increase with increasing force amplitude.…”

“…Next, the mechanics of the flapping insect wing hinge are not fully understood and require further investigation. As discussed previously, it is possible that the transmission between thorax and wing is dynamic and contains a series-elastic element, though series elastic effects appear to be variable between insect species [ 19 ]. Additional research is required to characterize the behavior of the wing hinge so that it can be incorporated into mathematical models.…”

“…While many mathematical modeling efforts to date have focused on individual components (e.g., the isolated wing or thorax) rather than the system-level dynamics, some have formulated simple single-degree-of-freedom (SDOF) models to investigate coupling between the thorax, wing and surrounding fluid environment [ 18 , 19 ]. In the simplest configuration, a rigid wing and thorax are coupled via a static linear transmission, where a prescribed amount of thorax displacement corresponds to a certain amount of wing rotation.…”

“…For example, an SDOF model and accompanying robophysical experiment showed that the resonant frequency of the wing–thorax system was dependent on wing rotation amplitude due to nonlinear quadratic fluid damping [ 18 ]; absent nonlinear damping, the resonant frequency depended only on the mass and stiffness of the wing and thorax. SDOF models have also demonstrated that flying insects can achieve energy savings over a broad frequency range, whereas it was previously believed that optimal energetic performance was centered narrowly around the system’s resonant frequency [ 19 ].…”

“…While the derived model cannot describe aerodynamic lift, it provides a foundation to better assess energy flow through a multi-body system with fluid dissipation. Non-lifting models are simple yet useful tools for investigating the physics of flapping flight and are common in precedent work [ 18 , 19 ]. Insights garnered from this model can inform the compliant design of bio-inspired FWMAVs and lay a foundation for more complex models that accommodate MDOF rotations and lifting flows.…”