Hydrodynamic loading of microcantilevers oscillating near rigid walls

The vibrations of microcantilevers in atomic force microscopes (AFMs) or radio frequency (RF) switches are strongly influenced by the viscous hydrodynamics of the surrounding fluid in the vicinity of a rigid wall. While prior efforts to model this hydrodynamic loading have focused on squeeze film damping effects at high Knudsen and squeeze numbers, the regimes of low Knudsen and squeeze numbers are also very important for which squeeze film models need to be discarded in favor of unsteady Stokes hydrodynamics. We extend the work of Green and Sader [Phys Fluids 17, 073102 (2005); J. Appl. Phys. 98, 114913 (2005)] and present compact semianalytical formulas for the unsteady viscous hydrodynamic function of slender microbeams oscillating near rigid walls, in terms of key nondimensional numbers. Using these closed-form expressions, it becomes possible to predict easily the wet natural frequencies and quality factors of multiple modes of microcantilevers near rigid walls in diverse applications ranging from AFM in liquids to RF microswitches under ambient conditions. The semianalytical formulas are extensively validated by comparing their predicted wet natural frequencies and quality factors with those based on three-dimensional, transient flow-structure interaction simulations, as well as previous experiments performed in the field by other researchers. (C) 2008 American Institute of Physics. [DOI: 10.1063/1.3033499]