Analytical solution of flexural vibration responses on taped atomic force microscope cantilevers

An analytical solution of flexural vibration responses on taped atomic force microscope (AFM) cantilevers has been derived and a closed-form expression obtained. The results coincide with previously published analytical solutions for two special cases that are used with the rectangular AFM cantilever. The analytical solution derived in this Letter, can easily assist in obtaining the resonance frequency at arbitrary dimensions and tip radii. It is useful for the design of the AFM cantilever, including the solid or hollow circular and rectangular cross-section, and can serve to evaluate the accuracy of the approximate or numerical solutions for the AFM cantilever with a complicated cross-section. Furthermore, the solution was confirmed and can also be applied to obtain the resonant frequency of a conical cantilever with an elongated tip which is used for imaging the deep trenches and complex structures