Oscillations of a beam in fluid flow : a means for underwater propulsion

Aquatic animals commonly oscillate their fins, tails, or other structures to propel and control themselves in water. These elements are not perfectly rigid, so the interplay between their stiffness and the fluid loading dictates their dynamics. We examine the propulsive qualities of a tail-like flexible beam over a range of flow speeds, with oscillations induced either by a known forcing function, feedback-based actuation, or a follower force. This is accomplished using the equations of fluid-immersed beams in combination with a set of tractable expressions for thrust and efficiency. We first compute these equations for a flexible beam actuated by a known forcing function over the external flow velocity and forcing frequency plane and show that the flexible propulsor has regions of both positive and negative thrust. We show the behavior of a sample underwater vehicle with fixed drag characteristics as an illustration of a realizable system.An alternate approach to generating the tail motion for underwater propulsion is to use a method where the oscillation of the flexible element is self-induced. We investigate a pinned-free beam in axial fluid flow, subjected to feedback-based actuation at the pinned end. The actuation may be a moment or a prescribed angle, and it is proportional to the state (curvature, slope, or displacement) of the beam at some point along its length. All equations and boundary condition terms are non-dimensionalized and the stability of the system is studied over a range of external flow velocities and sensing locations. For each combination of flow velocity and sensing location, the critical gain (positive or negative) for the onset of flutter is determined. This process, which is repeated for each combination of actuation and sensing modes, reveals that the closed-loop system exhibits a rich set of stability transitions, each associated with a traveling waveform in the flexible beam at the onset of flutter. With the intent of exploring the use of flexible fluttering beams for underwater propulsion, the efficiency of these waveforms is computed using slender-body theory. Additional insights into the efficiency of the waveforms are obtained through considerations of the smoothness of the traveling waveforms. Using a water tunnel at various flow speeds, we provide experimental validation of feedback-induced flutter of a beam for the specific case of moment actuation proportional to curvature. We show that the model, adapted for experimental considerations, results in flutter at very similar frequencies as the experimental results.We complete our investigation of flutter-based propulsion by examining how the onset of flutter is affected by the application of a follower force to a cantilevered beam in fluid flow. We consider the full range of fluid-mass to beam-mass ratios (from thick pipes to thin beams) given the possibility that the follower force could be generated by internal flow and a fluid jet. We follow the stability of the system as the flow velocity is increased to determine the onset of flutter. Over this full range of mass ratios and external flow velocities, we determine the critical follower force and critical frequency. This set of investigations, along with our investigations of forced oscillations and feedback-induced flutter, provide insights into how a flexible propulsor may be used as an alternative to the propeller.