Resonant frequencies of a rectangular cantilever beam immersed in a fluid

The resonant frequencies of cantilever beams can depend strongly on the fluid in which they are immersed. In this article, we expand on the method of Elmer and Dreier [J. Appl. Phys. 81, 7709 (1997)] and derive explicit analytical formulas for the flexural and torsional resonant frequencies of a rectangular cantilever beam immersed in an inviscid fluid. These results are directly applicable to cantilever beams of macroscopic size, where the effects of viscosity are negligible, and are valid for arbitrary mode number. In contrast to low mode numbers, in all cases it is found that the fluid has no effect on the resonant frequencies in the limit of infinite mode number.