Resonant frequencies of cantilevered sheets under various clamping configurations immersed in fluid

Immersion of an elastic cantilevered sheet in a fluid can strongly affect its dynamic response. While significant effort has been expended in studying slender cantilevered sheets, the behavior of wide sheets has received far less attention. Here we study the clamping configuration's effect on the vibrational dynamics of wide cantilever sheets of macroscopic size, which naturally generate inviscid flows. Three practically relevant clamping configurations are investigated: clamping into (i) a thin and rigid horizontal plate, (ii) a rigid vertical wall, and (iii) a rigid line. These are found to produce different resonant frequencies, as expected from the nonlocal flows generated by these cantilevers. The resulting formulas are joined to an existing expression for slender cantilevers, leading to a universal formula valid for all aspect ratios (cantilever length/width) and mode numbers; accuracy is verified using finite element analysis. This study is expected to be of practical value in a host of engineering applications, such as those that utilize fluid-structure interactions for energy harvesting and aerodynamic design.