Torsional frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope

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Abstract
The frequency response of a cantilever beam is strongly dependent on the fluid in which it is immersed. In a companion study, Sader [J. Appl. Phys. 84, 64, (1998)] presented a theoretical model for the flexural vibrational response of a cantilever beam, that is immersed in a viscous fluid, and excited by an arbitrary driving force. Due to its relevance to applications of the atomic force microscope (AFM), we extend the analysis of Sader to the related problem of torsional vibrations, and also consider the special case where the cantilever is excited by a thermal driving force. Since longitudinal deformations of AFM cantilevers are not measured normally, combination of the present theoretical model and that of the companion study enables the complete vibrational response of an AFM cantilever beam, that is immersed in a viscous fluid, to be calculated.