This paper investigates stability analysis of flapping flight. Due to time-varying aerodynamic forces, such systems do not display fixed points of equilibrium. The problem is therefore approached via a limit cycle analysis based on Floquet theory. Stability is assessed from the eigenvalues of the Jacobian matrix associated with the limit cycle, also known as the Floquet multipliers. We developed this framework to analyze the flapping flight equations of motion of a bird in the longitudinal plane. Such a system is known to be not only nonlinear and time dependent, but also driven by state-dependent forcing aerodynamic forces. A model accounting for wing morphing under prescribed kinematics is developed for generating realistic state-dependent aerodynamic forces. The morphing wing geometry results from the envelope of continuously articulated rigid bodies, modeling bones and feather rachises, and capturing biologically relevant degrees of freedom. A sensitivity analysis is carried out which allows studying several flight configurations in trimmed state. Our numerical results show that in such a system one instability mode is ubiquitous, thus suggesting the importance of sensory feedback to achieve steady-state flapping flight in birds. The effect of wingbeat amplitude, governed by the shoulder joint, is found to be crucial in tuning the gait toward level flight, but marginally affects stability. In contrast, the relative position between the wing and the center of mass is found to significantly affect the values of Floquet multipliers, suggesting that the distribution of pitching moment plays a very important role in flapping flight stability.

本文研究了扑翼飞行的稳定性分析。由于随时间变化的空气动力，这种系统没有显示出固定的平衡点。因此，该问题是通过基于Floquet理论的极限循环分析来解决的。根据与极限循环（也称为浮球乘数）相关联的雅可比矩阵的特征值来评估稳定性。我们开发了这个框架来分析鸟类在纵向平面上的拍打飞行方程。已知这样的系统不仅是非线性的和时间相关的，而且是由状态相关的强迫空气动力驱动的。开发了一种在规定的运动学下考虑机翼变形的模型，用于生成逼真的状态相关的空气动力。不断变化的机翼几何形状是由连续铰接的刚体的外壳，骨骼和羽毛轴的建模以及捕获生物学上相关的自由度产生的。进行了灵敏度分析，可以研究处于修剪状态的几种飞行配置。我们的数值结果表明，在这样的系统中，一种不稳定性模式是普遍存在的，因此表明感觉反馈对于实现鸟类的稳态扑动飞行的重要性。人们发现，由肩部关节控制的翼拍振幅的影响对于将步态调向水平飞行至关重要，但对稳定性影响不大。相反，发现机翼和质心之间的相对位置会显着影响Floquet乘数的值，