Three-dimensional self-propelled flexible plate with time-varying flapping frequency

The variation in the flapping frequency of a self-propelled flexible plate over time is analysed, and its transient propulsion is examined using the immersed boundary method. The flapping frequency of the flexible plate is continuously defined as a function of time using a piecewise definition. Inspired by the similarity between the time-dependent variation of the average cruising speed and a step response plot, the settling time and maximum overshoot of behavior of the flexible plate are defined and scrutinized. By controlling the rate of change in the flapping frequency, the power consumption of the flexible plate is optimized. As a result, the flapping frequency of the flexible plate gradually transitions from the most efficient flapping frequency to the flapping frequency that achieves the highest average cruising speed. During this transition, the power consumption of the flexible propulsor is reduced to 1/4 of its original value, while the settling time decreases to approximately 38% of its initial duration. To analyse the propulsion mechanisms of the flexible propulsor from the perspective of vortex dynamics, vortical structures are identified through percolation theory. To investigate the influence of the identified vortical structures on the propulsion mechanisms of the flexible plate, an inverse power law-based formulation is proposed and validated by comparing it with the time-dependent propulsion speed and the power consumption of the flexible propulsor. Such transient propulsion is commonly observed in various operational environments, such as acceleration and deceleration phases of unmanned underwater vehicles (UUVs) and flapping-wing air vehicles (FWAVs). The present work is expected to serve as a foundational investigation for understanding oscillatory motion under these conditions.