Time-Periodic Stability of a Flapping Insect Wing Structure in Hover

The wings of insectlike flapping-wing micro air vehicles experience time-periodic inertial stiffnesses during flapping motion. By modeling the wing structure as a thin beam, a linear time-periodic assumed-modes analysis is developed. The equations of motion of a flapping wing undergoing out-of-plane bending and torsion are derived. The homogeneous assumed-modes equations are nondimensionalized. It is shown that the nondimensional strain stiffness varies with the ratio of the wing's nonrotating natural frequencies to the flapping frequency, whereas the nondimensional, time-periodic inertial stiffnesses vary with the amplitudes of flapping and feathering motion. The parametric stability of a representative wing is assessed by applying Floquet analysis to the nondimensional equations of motion, and a scalable stability diagram is presented. Parametric instabilities of the wing structure, caused by time-periodic stiffnesses, are characterized and plotted in the time domain. The effects of important structural design properties on parametric stability are examined.