Floquet stability analysis of the longitudinal dynamics of two hovering model insects

Abstract

Because of the periodically varying aerodynamic and inertial forces of the flapping wings, a hovering or constant-speed flying insect is a cyclically forcing system, and, generally, the flight is not in a fixed-point equilibrium, but in a cyclic-motion equilibrium. Current stability theory of insect flight is based on the averaged model and treats the flight as a fixed-point equilibrium. In the present study, we treated the flight as a cyclic-motion equilibrium and used the Floquet theory to analyse the longitudinal stability of insect flight. Two hovering model insects were considered-a dronefly and a hawkmoth. The former had relatively high wingbeat frequency and small wing-mass to body-mass ratio, and hence very small amplitude of body oscillation; while the latter had relatively low wingbeat frequency and large wing-mass to body-mass ratio, and hence relatively large amplitude of body oscillation. For comparison, analysis using the averaged-model theory (fixed-point stability analysis) was also made. Results of both the cyclic-motion stability analysis and the fixed-point stability analysis were tested by numerical simulation using complete equations of motion coupled with the Navier -Stokes equations. The Floquet theory (cyclic-motion stability analysis) agreed well with the simulation for both the model dronefly and the model hawkmoth; but the averaged-model theory gave good results only for the dronefly. Thus, for an insect with relatively large body oscillation at wingbeat frequency, cyclic-motion stability analysis is required, and for their control analysis, the existing well-developed control theories for systems of fixed-point equilibrium are no longer applicable and new methods that take the cyclic variation of the flight dynamics into account are needed.

Nonlinear flight dynamics and stability of hovering model insects

“…It is expected that for small disturbance motion, the numerical solution is close to the analytical one. For the hovering hawkmoth and dronefly, the analytical solution of small disturbance motion was obtained by our group [11][12][13]: …”

“…With the averaged model, the standard aircraft equations of motion [10] can be used for flying insects. Recent numerical [11,12] and theoretical [3,13] studies have shown that the averaged model works very well for insects who have relatively high wingbeat frequency and small wing-mass to body-mass ratio (hence very small amplitude of body oscillation); for insects who have relatively low wingbeat frequency and large wing-mass to body-mass ratio (hence relatively large amplitude of body oscillation), the averaged model works less well, nevertheless, it can still correctly predict the trend of variation of the dynamic properties. The linear theory assumes that the animal's motion consists of small disturbances from the equilibrium flight; thus, the equations of motion are linearized about the equilibrium point, and the aerodynamic forces and moments are represented as analytical functions of the motion variables (state variables) and the aerodynamic derivatives [10].…”

Dynamics and flight control of a flapping-wing robotic insect in the presence of wind gusts

“…On the other hand, in the context of flight stability, researchers have attempted to quantify changes in lift and drag on flapping wings in forward [35][36][37][38]40] and lateral flight [35,39]. In two studies [35,36], researchers have proposed that the wings act as a source of drag that is linear in body velocity or air speed owing to the averaged drag on two half-strokes.…”

“…The effects of wind disturbances on flapping flight are different from fixed-wing and rotary-wing flight in nature owing to unsteady aerodynamics. Literature related to wind gusts and flapping flight could be categorized into either the study of flapping-wing aerodynamics in the presence of gusts or turbulence [30][31][32][33][34] or the study of stability in forward and lateral flight [35][36][37][38][39][40]. Research focusing on aerodynamics typically employs computational fluid dynamics (CFD) or flow visualization to quantify instantaneous (as opposed to stroke averaged) flow around the wings and deduce corresponding aerodynamic properties such as lift and drag coefficients.…”

Wing in the Loop: Integrating the Wing into Dynamics of Insect Flight

“…For instance, similar to the conventional flight dynamics techniques, stability of the insect flight is commonly investigated by linearizing the equations of motion about a specific flight state (usually equilibrium). These equations can be solved by Floquet theory [Nayfeh andMook, 2008, Wu andSun, 2012] or by numerical simulations [Taylor and Zbikowski, 2005].…”