  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 to 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 non-linear 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 towards 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.

Fluid Dynamics;Chaotic Dynamics