Aquatic locomotion by a flapping tail with passive pitch and tunable stiffness

Aquatic locomotion by a flapping tail, typical of thunniform fish species, is commonly involving a prescribed heaving motion and a passive pitch resulting from the interaction with the surrounding fluid. A very simple model of a fishlike body can be generated by a flat plate connected via a torsional spring to a virtual body carrying the total integral quantities of mass and resistance. To simulate real fish, a tunable stiffness of the spring is essential to reproduce the swimming behaviour of these species in nature. All the above aspects of the model allow for approximate analytical solutions which are suitable for easily exploring the entire parameter space to search the best performance conditions. In addition, they may be of great help for applications in the framework of control theory. A realistic representation follows, and an interesting solution of the stride length as a function of a non-dimensional frequency is discussed. A favorable effect of the resonance region appears, showing also how the stiffness should be modified in nature, by tuning the muscular tension to maintain optimal locomotion for a given change of frequency. For a biomimetic robot, instead, a proper tool may be implemented to this purpose. The location of the maxima for the relevant output quantities is explained by a novel physical interpretation which has been obtained through a parallel investigation of the same body configuration under an oncoming uniform stream with different velocity values. The analysis identifies the role of fluid damping to shift the location of the maxima for the different quantities considered in the discussion of the results.