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We have reported that how base modes alternate with
cantilever modes both in our model and in the FEM model,
see Figure 6 (a). The relative errors between both
eigenvalues sequences are represented in Figure 6 (b). Note
that errors are far from being uniform among eigenvalues. In
fact, the main error source resides in a poor precision of the
beam model for representing base deformations in some
particular deformation modes.
Fig. 6. Eigenmode density distributions for finite element model and
for the two-scale model
VI. CONCLUSION
A two-scale model for two-dimensional cantilever arrays
in dynamic regime has been derived based on a theory of
strongly heterogeneous homogenization where the
cantilevers play the role of soft parts. We conclude to a
globally good agreement with the three-dimensional
elasticity model based on eigenvalue density and mode shape
comparisons. The validation of the model demonstrates that
the two-scale model was sufficiently light to apply to
two-dimensional AFM arrays. More comparisons with FEM
results are still needed for large arrays.
ACKNOWLEDGMENT
This work is partially supported by the European
Territorial Cooperation Programme INTERREG IV A
France-Switzerland 2007-2013. The Computations have
been performed on the super computer facilities of the
Mésocentre de calcul de Franche-Comté.
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