Modeling the fluid-structure interaction of micro-plate-resonators in viscous fluids

Abstract

Cantilevered thin structures are among the most common building block of micro-resonators. In fluidic environments, the surrounding fluid dissipates energy from the micro-resonator. The underlying viscous fluid-structure interaction is well understood for slender resonator geometries. However, for non-slender resonator geometries determining fluid losses remains challenging with both analytical and numerical approaches.Here, we present a semi-numerical method for determining the steady-state dynamics of wide micro-plate resonators in viscous fluids. The method is based on the Kirchhoff plate equation to solve for the plate’s dynamics, while the hydrodynamic force acting on the plate is determined from the Stokes equations with a boundary integral formulation. The boundary integral formulation avoids discretizing the entire fluid domain and thus avoids multi-scale issues. Two fluid flow formulations are introduced here, the first in which a two-dimensional fluid flow is assumed, and the second formulation allows fora three-dimensional fluid flow. The equation of motion is solved with the finite element method. In numerical examples, the method is convergent with an exponent of the convergence rate equal to 2. We determine quality factors of micro-plates in liquids and observe excellent agreement with experimental data. Since the proposed method goes beyond existing semi-analytical methods by accounting for two-dimensional vibrational modes, novel unseen effects are investigated. For instance, in gases, the Euler-Bernoulli(EB) modes (modes with nodal lines only along the plate’s width) exhibit the lowest Q-factors, while non-EB modes exhibit the highest Q-factors. The opposite is found in liquids, as EB modes show the highest Q-factors, and non-EB modes lower Q-factors.We name this opposite Q-factor paern in gases and liquids the gas-liquid-Q-inversion(GL-Q-inversion). Experiments in water and air showed a Q-factor agreement with the GL-Q-inversion, and differences in Q-factor between simulation and experiments were below 25%. Differences in the resonance frequency are high for the EB modes in water due to the two-dimensional fluid flow approximation. The second method is proposed,in which a three-dimensional fluid flow is investigated using the unsteady Stokeslet. Results with the 3D fluid flow method exhibit even beer agreement between simulation and experiments. The results and methods shown here will pave the way to efficiently exploit the two-dimensional vibrational modes of non-slender resonators to improve MEMS performance in gaseous and liquid environments.