A numerical method to determine the displacement spectrum of micro-plates in viscous fluids

Abstract

In this paper, we present a semi-numerical method for determining the dynamics of micro-resonators with finite width immersed in incompressible viscous fluids. The micro-resonator is modeled using Kirchhoff plate theory, and the hydrodynamic force acting on the plate is determined from a boundary integral formulation of the Stokes equations. The resulting equation of motion is solved with a continuous/discontinuous finite element method in which an interior penalty term imposes C1-continuity to the plate’s deflection. Numerical investigations show the method to be convergent with an exponent of the convergence rate equals 2. Examples demonstrate that the proposed method overcomes the limitations of existing semi-analytic methods, only applicable to beam geometries, considering arbitrary plate modes in the structure’s dynamics and their effects on the fluid flow. Different resonator geometries are investigated for which displacement spectrum, mode shapes, and quality factors are not determinable with existent semi-analytic methods. Moreover, we find excellent agreement between simulated and experimental data, which has not been achieved even with purely numerical methods. The present method will allow the understanding of high quality factors of wide micro-resonators in viscous fluids and facilitate new applications in liquid atomic force microscopy and gas sensing in ambient and low-pressure conditions.