Frequency response of cantilevered plates of small aspect ratio immersed in viscous fluids

Abstract

Comprehensive theoretical models for the dynamic response of slender cantilevered beams immersed in fluid have been widely reported, while the distinct behavior of wide cantilevered plates has received comparatively little attention. In this article, we develop an exact analytical theory for the resonant response of rectangular cantilevered plates of zero length-to-width aspect ratio that are immersed in unbounded viscous fluids. Unlike the opposite slender limit of large aspect ratio, the hydrodynamic load experienced by zero-aspect-ratio cantilevered plates is inherently non-local, which can strongly affect the individual mode shapes of the plate. In addition, finite-element-method simulations are reported for two- and three-dimensional cases of zero and finite aspect ratio, respectively. Accuracy of the present theory and that of Atkinson and Manrique de Lara [J. Sound Vib. 300, 352 (2007)] for small viscosity and zero aspect ratio is assessed using the former simulations. The latter simulations are used to clarify the regime of validity of the present theory as a function of aspect ratio, along with that of existing theory for slender (large aspect ratio) beams. The results of this study are expected to be of practical importance to micro- and nano-electromechanical system design and their applications.