A new theoretical model is presented for an aquatic vehicle self-propelled by a rigid foil undergoing pitching oscillations generated by a torque of small amplitude applied at an arbitrary pivot axis at which the foil is elastically supported to allow for passive heaving motion. The model is based on 2D linear potential-flow theory coupled with the self-propelled dynamics of the semi-passive flapping foil elastically mounted on the vehicle hull through translational and torsional springs and dampers. It is governed by just three ordinary differential equations, whose numerical solutions are assessed with full viscous numerical simulations of the self-propelled foil. Analytical approximate solutions for the combined effect of all the relevant non-dimensional parameters on the swimming velocity and efficiency are also obtained by taking advantage of the small-amplitude of the applied torque. Thus, simple power laws for the velocity and efficiency dependencies on Lighthill number and torque intensity are obtained. It is found that the swimming velocity and transport efficiency can be greatly enhanced by selecting appropriately the non-dimensional constants of the translational and torsional springs, which are mapped for typical values of the remaining parameters in aquatic locomotion. These resonant values serve to select optimal frequencies of the forcing torque for given structural and geometric parameters. Thus, the present model and analysis provide a useful guide for the design of an efficient flapping-foil underwater vehicle.
