Hydrodynamic forces in non-uniform cantilever beam resonator

In this paper, we developed two dimensional and three dimensional boundary element method (BEM) to compute hydrodynamic forces due to the oscillation of non-uniform beam (NUB) in a quiescent incompressible fluid with linear and quartic varying widths. To model the fluid flow under small amplitude oscillation of thin NUB in its first mode, the linearized unsteady Stokes equation is solved using BEM. After finding the converged structural and fluid nodes in all the cases, we compute real and imaginary components of hydrodynamic function. Subsequently, damping ratio or quality factor is found from energy dissipation due to drag forces mainly because of stress jumps across the thin beam thickness. Similarly, the frequency shift is found due to virtual added mass obtained from the mean hydrodynamic thrust force. The results are validated with existing literature and further analysis is done in terms of tapering parameter and index of non-uniform beam, and the corresponding aspect ratio and frequency parameters. Based on the analysis presented, it is found that quartic converging beam provides better quality factor and least added mass effect and it can be explored to design a cantilever based resonator operating in fluid with improved performance such as AFM probes. Thus, the new model developed for non-uniform beam can be useful to drag forces in other types of 2D and 3D beams.