School of Transportation Science and Engineering, Beihang University, Beijing, People's Republic of China.
J R Soc Interface. 2012 Sep 7;9(74):2033-46. doi: 10.1098/rsif.2012.0072. Epub 2012 Apr 4.
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and for their control analysis, the existing well-developed control theories for systems of fixed-point equilibrium are no longer applicable and new methods that take the cyclic variation of the flight dynamics into account are needed.
由于扑翼的空气动力和惯性力是周期性变化的，悬停或定速飞行的昆虫是一个周期性的力系统，通常，飞行不是在固定平衡点，而是在周期性运动平衡点。昆虫飞行的当前稳定性理论基于平均模型，将飞行视为固定平衡点。在本研究中，我们将飞行视为周期性运动平衡点，并使用 Floquet 理论分析昆虫飞行的纵向稳定性。考虑了两种悬停模型昆虫——蝇和天蛾。前者具有较高的翅膀拍打频率和较小的翅膀质量与身体质量比，因此身体摆动幅度很小；而后者具有较低的翅膀拍打频率和较大的翅膀质量与身体质量比，因此身体摆动幅度相对较大。为了进行比较，还使用平均模型理论（固定点稳定性分析）进行了分析。使用完整的运动方程和纳维-斯托克斯方程耦合的数值模拟对周期性运动稳定性分析和固定点稳定性分析的结果进行了测试。对于模型蝇和模型天蛾，Floquet 理论（周期性运动稳定性分析）与模拟结果吻合良好；但是，平均模型理论仅对蝇的结果有效。因此，对于在翅膀拍打频率下具有相对较大身体摆动的昆虫，需要进行周期性运动稳定性分析，并且对于它们的控制分析，现有的针对固定平衡点系统的成熟控制理论不再适用，需要考虑飞行动力学的周期性变化的新方法。
