_applied sciences_  
_Article_  
Three-Dimensional Hydrodynamic Analysis of a Flexible  
Caudal Fin  
May [PERSON] 1,2,*,+\({}^{\dagger}\) and [PERSON] 2,*,+\({}^{\ddagger}\)  
resonance of an elastic plate and its deformation. An experimental investigation on the propulsive performance of a heaving flexible panel showed that the time-averaged net thrust experiences localized boosts near resonant frequencies where the trailing edge amplitude is maximized [12]. [PERSON] et al. [13] experimentally investigated the influence of passive tail flexibility on thrust production using a robotic lamprey. They reported that the force production could be increased when the cycle frequency was changed to the resonant frequency of the tail. A freely moving wing with active heaving in a fluid was studied experimentally and numerically by [PERSON] et al. [14]. They showed that the horizontal velocity was largest when the flapping frequency was near the immersed system's resonant frequency. The computations of [PERSON] et al. [15] presented the role of resonance in swimming performance by using a three-dimensional flexible panel heaving sinusoidally at its leading edge. This study showed that the local peaks in the normalized trailing edge amplitude achieved by the panel coincide with the local peaks in forward swimming speed. The work of [PERSON] et al. [16] focused on a simplified two-dimensional propulsion problem using a flexible wing of infinite span, actuated at its leading edge in a purely heaving motion. Their results indicated that the resonant frequency could induce maximum values of the mean developed thrust and power input. In the analytical work of [17], the propulsive performances were investigated using a two-dimensional foil with pitching and heaving motions about arbitrary pivot points. Their results also indicated that the local maxima in the amplitude of the flexural component of the deflection roughly correspond to local maxima in thrust and in power coefficients. However, results from other studies in self-propelled conditions [19; 20; 21] reported that the optimal flapping frequency for the maximum cruising velocity and maximum propulsive efficiency appeared at a frequency below the resonance. [PERSON] et al. [19] conducted an experimental investigation on the performance of flapping flyers with a self-propelled simplified insect model in air. Their results indicated that maximum performance was obtained at a frequency lower than the resonance. This observation coincides with the results reported by [PERSON] et al. [20] for a self-propelled model of flexible wing flapping in air, in which the optimum flapping frequency for the propulsive efficiency occurs below the natural relaxation frequency. Numerical simulation of a self-propelled plunging foil was conducted by [PERSON] et al. [21] to investigate the hydrodynamic effect of passive flexibility. They showed that the optimum performance is always achieved at a forcing frequency much lower than the resonance point. On the other hand, [PERSON] et al. [22] studied numerically the thrust generation of a fixed flexible body in air and reported that the propulsive efficiency reached a maximum at a certain flapping frequency above the resonant frequency. [PERSON]'s experiments [23] on a flexible panel oscillating in water with different elastic modulus showed that the structural resonance could not contribute to the efficient locomotion of flexible propulsors. This efficient locomotion was predicted to occur by considering both the optimum Strouhal number and structural resonant frequency. In the experimental work of [PERSON] et al. [24], it was found that the thrust developed by the flexible robotic fish depend on the amplitude of tail and oscillation frequency, irrespective of whether the oscillation frequency is resonance or not. On the other hand, the mechanism of thrust generation in flapping-type propulsion is still a matter of debate.