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A. Gesing, D. Platz, U. Schmid:
"Modeling non-conventional vibrational modes of micro-resonators in viscous fluids";
Talk: ICoNSoM 2019 - International Conference on Nonlinear Solid Mechanics, Rom, I; 06-20-2019 - 06-23-2019; in: "International Conference on Nonlinear Solid Mechanics", (2019), 1.

Micro-cantilevers are among the most common structures used as micro-resonators in applications
such as high-sensitivity mass measurements and atomic force microscopy (AFM). Conventionally
only 1D vibrational modes of micro-cantilevers are considered (Fig. 1a), and methods
to predict the cantilever´s dynamic response in viscous fluids are well established. Recently
interest sparked in using the 2D non-conventional vibrational modes (Fig. 1b) of micro-plates
in novel micro-resonators. Of particular interest are these structures´ proven extraordinarily
high quality factors at low frequencies in liquid media. There is therefore an increased demand
for a method that predicts the dynamic response of micro-plates in viscous fluids. Two
major difficulties arise when considering 2D vibrational modes of plates in fluids. First, the
commonly used Euler-Bernoulli theory describes only 1D vibrational modes. To model 2D
modes, Kirchhoff-Love plate theory (KLPT) is required. However, for cantilevered boundary
conditions no analytic solution is known. In addition, KLPT is based on a fourth order
partial differential equation, what impedes its solution with standard finite element method.
Second, the viscous force acting on the 1D micro-cantilever is local and linearly dependent
on its displacement. In micro-plates oscillating in non-conventional modes the viscous force
is, in contrast, non-linear and non-local in relation to the displacement. In this work we solve
the Kirchhoff-Love plate equations with a stabilized interior penalty formulation of continuous/
discontinuous Galerkin method. This method enables us to use Lagrange-type continuous
elements while weakly enforcing boundary conditions and penalizing discontinuities in higher
order derivatives. The fluid flow is modeled with linearized Navier-Stokes equations, and solved
with 2D Green´s functions. In Figs. 1c and 1d we present the flow fields around a plate due to
conventional and non-conventional vibrational modes respectively. The vortices near the edges
are considerably smaller in amplitude and size in the non-conventional mode flow field, which
explains this mode´s higher quality factor. The non-linear fluid-structure coupling is computed
with Jacobian-free Krylov solvers. Numerical results are compared with published experimental
data. We anticipate that the presented method will greatly impact the design of micro-resonators
and AFM by enabling the prediction of the dynamic response of novel devices.