Online proceedings - EDA Publishing Association

11-13
May 2011, Aix-en-Provence, France
A Model for Two-Dimensional Arrays of Cantilevers
in the Dynamic Regime
Hui HUI 1, 2 , Michel LENCZNER 2
1 School of Mechatronic Northwestern Polytechnical University,
127, Youyi Xilu,
710072 Xi’an Shaanxi, China
2 FEMTO-ST, Département Temps-Fréquences Université de Franche-Comté,
26, Chemin de l’Epitaphe,
Abstract- We present a model for two-dimensional arrays of
micro-cantilevers in elasto-dynamical regime. It has been
derived by a two-scale approximation method related to
strongly heterogeneous system. We also report validation
results regarding its modal structure compared with the one of
a direct Finite Element Model (FEM).
I. INTRODUCTION
Since its invention [1], the Atomic Force Microscope
(AFM) has open new directions for a number of operations at
the nanoscale with an impact in various sciences and
technologies. A number of research laboratories are now
developing large arrays of AFM that can achieve the same
kind of task in parallel. The most advanced system is the
Millipede from IBM [2] for data storage, but again, a number
of new architectures are emerging see [3], [4], [5], [6], [7].
The main limitations of the AFM devices are their low
speed of operation and their low reliability, this is even more
true for arrays. Thus, the modeling of single AFM and their
model-based control are more and more studied, see M.
Napoli et al. [8], S.M. Salapaka et al. [9], M. Sitti [10] for
instance. Regarding arrays, only the group of B. Bamieh, see
[8] and the reference therein, has published a model of
coupled cantilever arrays. It takes into account an
electrostatic coupling, and its derivation is
phenomenological. It turns out that numerical simulations
must handle the full array, leading to a prohibitive
computational time for the time scale of a designer.
The goal of this paper is to present a simplified model for
the elastic behavior of large two-dimensional cantilever
arrays as these appearing in AFM arrays, as depicted in
Figure 1. It extends the paper [11] by taking into account the
dynamical regime instead of the static regime, and it applies
to two-dimensional arrays instead of one-dimensional arrays.
A detailed paper has been submitted for publication, it
includes all necessary information about the model and its
derivation. The corresponding model for one-dimensional
arrays in dynamic regime was announced in the letter [12].
Our model is mainly based on a homogenization technique
dedicated to strongly heterogeneous materials or systems
expressed in the framework of two-scale convergence (or
approximation) as introduced in M. Lenczner [13], [14] or in
25030 Besançon Cedex, France
D. Cioranesco, A. Damlamian and G. Griso [15]. In a
preliminary step, its derivation also make use of the
asymptotic method related to thin structures of P.G. Ciarlet
[16] and of P. Destuynder [17]. We emphasize that the choice
of a method for the modeling of the periodic array is not
straightforward, in particular, a standard homogenization
method is not relevant. On this point of view, a particular
feature of a cantilever array is that the local mechanical
displacements of the moving parts may be of the same order
of magnitude as the displacements of the common support.
Another point is that the lowest local eigenfrequencies of the
moving parts are also in the same order of magnitude as those
of the common supporting base. These features may be usual
in many microsystems arrays but not in continuum
mechanics for which the homogenization methods were
developed. So the usual homogenization methods, do not
yield interesting models even with introduction of correctors.
We review the main features of our simplified model. The
array is comprised of cantilevers clamped in a common base,
and possibly being equipped with tips. We assume that the
base is much stiffer than the cantilevers. This is expressed by
saying that their stiffness have different asymptotic
behaviors. The resulting model is composed of two evolution
equations, one for the macroscopic behavior, related to the
supporting base, and the other one at the microscopic level,
which takes into account the cantilever dynamics. As
required, their time scales are in the same range of magnitude
and so is their mechanical displacements. We further assume
that the tip is perfectly rigid, this is a commonly accepted
assumption. All these assumptions yield our model with
which we have carried numerical simulations and
validations.
The paper is organized as follows. In Section II, we start
by describing the array geometry, and then shortly introduce
the two-scale approximation method. Then we introduce our
model in Section III. In Section IV, we discuss the
eigenmodes of the model and its validation is detailed in
Section V.
II. THE TWO-SCALE APPROXIMATION
We consider a two-dimensional array of cantilevers, see
Figure 1 (a) with cell represented in Figure 1 (b).
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