## A Scaling Parameter for Force Generation for Flexible Wings

The maximum propulsive force, such as thrust in forward flight or lift in hovering motion, was generated at a frequency that was slightly lower than the natural frequency of the system [222, 367, 453, 454, 494, 508] as shown in Section 4.4.2. Zhang, Liu, and Lu [222] using the lattice Boltzmann method, numerically studied a flexible flat plate modeled as a rigid plate with a torsional spring at the pivot point on the leading edge of the wing. They concluded that the flat plate would move forward and hence generate thrust when the leading edge plunges at a motion frequency that is lower than the natural frequency of the system; the flat plate would move backward if the frequency ratio, the ratio between the motion frequency and the natural frequency, is greater than one. Similarly, Masoud and Alexeev [508] used the lattice Boltzmann method to show that the maximal propulsive force is obtained at the frequency ratio of 0.95. The magnitude of the maximal force would increase when the inertial effects became more important than the fluid inertia. Michelin and Llewellyn Smith [494] used potential flow theory to describe the flow over a plunging flexible wing. They showed that the trailing-edge flapping amplitude and the propulsive force are maximal at resonance conditions. In a series of experiments using a self-propelled simplified insect model, Thiria and Godoy-Diana [453] and Ramana – narivo, Godoy-Diana, and Thiria [454] also showed that the maximum thrust force os around a frequency ratio of 0.7. More recently, Gogulapati and Friedmann [367] coupled an approximate aerodynamic model, which was extended to forward flight including the effects of fluid viscosity, to a non-linear structural dynamic model. For various setups of composite anisotropic Zimmerman wings [507], they investigated the propulsive force generation in forward flight. They found that the maximum propulsive force is also obtained at a frequency ratio slightly lower than one. These observations are consistent with the general perception of the resonance phenomena, in which even small external forces can induce large-amplitude deformations and potentially be efficient as well.

However, it has been reported for insects that the flapping frequency is below the natural frequencies of the wing and is only a fraction of the resonance frequency [509, 510] . Sunada, Zeng, and Kawachi [509] measured the natural frequencies of vibration in air and the wing-beat frequencies for four different dragonflies. The wing-beat frequency ratios were in the range of 0.30-0.46. Chen, Chen, and Chou [510] also measured the wing-beat frequencies and natural frequencies of dragonfly wings. In their measurements the average flapping frequency was 27 Hz, whereas the natural frequency, calculated using a spectrum analyzer, was 170 Hz when clamped at the wing base, resulting in a frequency ratio of about 0.16.

The propulsive efficiency was also investigated numerically [492,508] and experimentally using a self-propelled model [453] [455]. Vanella et al. [492] conducted numerical investigations on a two-link model and found that the optimal performance is realized when the wing is excited at a frequency ratio of 0.33. For all Reynolds numbers considered in the range of 7.5 x 101-1 x 103, the wake-capture mechanism is enhanced due to a stronger flow around the wing at stroke reversal, resulting from a stronger vortex at the trailing edge. Using the experimental setup described earlier, Thiria and Godoy-Diana [453] and Ramananarivo, Godoy-Diana, and Thiria [454] also showed that the maximum efficiency is obtained at a frequency ratio of 0.7, lower than that of the maximum propulsion. They concluded that the performance optimization is obtained not by looking at the resonance but by adjusting the temporal evolution of the wing shape. In contrast, Masoud and Alexeev [508] showed that the optimal efficiency for a hovering flat plate at Re = 100 occurs when the motion is excited at a frequency ratio of 1.25. In their setup a flexible flat plate had a geometric AoA of 40° in contrast to the previously mentioned studies where the plunging motion was symmetric.