International audienceDue to their low cost, size and precision M/NEMS are efficient sensors. M/NEMS sensors are used in various domains ranging from aeronautics to medicine or telecommunication, with applications such as chemical, inertial or mass sensing. Our previous researches on mass sensing were focused on a single resonator. In this work, a symmetric array of three resonant nanobeams is analysed. The originality lies in the use of a direct parametric analysis to sense an added mass during a symmetry-breaking event

Baguet, Sébastien

Dufour, Régis

Grenat, Clément

Lamarque, Claude-Henri

Nguyen, V-N.

English

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Mass detection through parametric analysis and symmetry-breaking in a MEMS array

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Mass detection through parametric analysis and
symmetry-breaking in a MEMS array
Clément Grenat, V-N. Nguyen, Sébastien Baguet, Régis Dufour, Claude-Henri
Lamarque
To cite this version:
Clément Grenat, V-N. Nguyen, Sébastien Baguet, Régis Dufour, Claude-Henri Lamarque. Mass de-
tection through parametric analysis and symmetry-breaking in a MEMS array. ENOC 2017, 9th
European Nonlinear Dynamics Conference, Jun 2017, Budapest, Hungary. ￿hal-01667934￿
ENOC 2017, June 25-30, 2017, Budapest, Hungary
Mass detection through parametric analysis and symmetry-breaking in a MEMS array
C. Grenat∗ , V-N. Nguyen∗, S. Baguet∗, R. Dufour∗ and C-H. Lamarque∗∗
∗ Univ Lyon, INSA-Lyon, CNRS UMR5259, LaMCoS, F-69621, Villeurbanne, France
∗∗Univ Lyon, ENTPE, CNRS, LTDS UMR5513, F-69518, Vaulx-en-Velin, France
Summary. Due to their low cost, size and precision M/NEMS are efficient sensors. M/NEMS sensors are used in various domains
ranging from aeronautics to medicine or telecommunication, with applications such as chemical, inertial or mass sensing. Our previous
researches on mass sensing were focused on a single resonator. In this work, a symmetric array of three resonant nanobeams is analysed.
The originality lies in the use of a direct parametric analysis to sense an added mass during a symmetry-breaking event.
Array of three nanomechanical resonators
Let a 3-beams-array be considered and sketched in Fig. 1. Each beam constitutes an electrostatic actuator for its adjacent
beams. All the beams share the same geometrical parameters and material properties: length l, width b, height h, Young’s
modulus E, moment of inertia I , material density ρ, gap g between two adjacent beams.
0 1 2 3
  nano-beam
Vdc10
Vac10
Vdc32
Vac32
g
h
w (x,t)s
x
O
Vdc21
Vac21
Vdc43
Vac43
4
Figure 1: Array of three clamped-clamped M/NEMS beams.
The sth beam is actuated by electrostatic forces generated by the adjacent beams. Vs,s+1 = Vdcs,s+1 + Vacs,s+1 cos(Ωt)
constitutes the voltage applied between the successive beams s and s + 1, with Vdc, Vac the continuous and alternative
voltage respectively. The resulting equations of motion take the following form [1] [2]
EI
∂4w˜s(x˜, t˜)
∂x˜4
+ ρbh
∂2w˜s(x˜, t˜)
∂t˜2
−
[
N˜ +
Ebh
2l
∫ l
0
(∂w˜s(x˜, t˜)
∂x˜
)2
dx˜
]∂2w˜s(x˜, t˜)
∂x˜2
=
0bCn
2
[
V 2s,s+1(
g + w˜s+1 − w˜s
)2 − V 2s−1,s(
g + w˜s − w˜s−1
)2
]
(1)
where s = 1, .., 3 and 0, Cn are the dielectric constant and fringing field coefficient respectively. N˜s represents the axial
preload along the x-axis set by design or due to manufacturing process. Beams 0, 4 on both extremities of the array are
totally clamped, the other beams 1, .., 3 are clamped-clamped. Therefore the boundary conditions are given by (2).
w˜0(x˜, t˜) = w˜n+1(x˜, t˜) = 0
w˜s(0, t) = w˜s(1, t) =
∂w˜s
∂x
(0, t) =
∂w˜s
∂x
(1, t) = 0 (2)
w˜0(x, t) = w˜n+1(x, t) = 0
Since the resonators of the 3-beams-array are identical, the beams 1, .., 3 have the same undamped linear eigenmodes.
Therefore these modes are used as a basis for the Galerkin method in order to remove the spatial dependence from the
equations of motion (1) and to obtain a reduced order model [3]. The resulting nonlinear differential system of equations
is then solved by means of the Harmonic Balance Method. The obtained periodic solutions are followed with the pseudo-
arc length method in order to obtain the response curves. Characterisation of bifurcations presented in [4] are then used to
detect bifurcation points such as Limit Points (LP) on the response curves. These bifurcations points can then be directly
followed to carry out a parametric analysis.
ENOC 2017, June 25-30, 2017, Budapest, Hungary
Mass Sensing by symmetry-breaking and Limit Points continuation
In order to use the symmetry-breaking event to detect and measure an added mass, the voltages are chosen as in Tab. 1.
Since the voltages are symmetric, the electrostatic forces acting on the central beam compensate each other. Thus the
central beam stays at rest until an added mass breaks the symmetry. When the symmetry-breaking event appears, the
central beam will start vibrating. For sufficiently high added mass or applied voltages, LP bifurcations will be present
within the dynamical response [4]. The induced hysteresis cycle is then used to detect the added mass [2].
V dc10 V ac10 V dc21 V ac21 V dc32 V ac32 V dc43 V ac43
0 0 5.3 1 5.3 1 0 0
Table 1: Voltage configuration of the three beams array.
In order to measure the added mass, a parametric analysis is performed. In Fig. 2a several responses curves for different
non-dimensional added masses have been computed, the non-dimensional mass m being defined as mp/ρbh with mp
the physical value of the added mass and ρbh the mass of one beam. However, this method is time consuming and is
not accurate. Instead of computing responses curves for several different added masses, it is more judicious to detect a
starting LP for a given value of the added mass (point 2 in Fig 2b) and then directly follow the branch of LP [4] with
respect to the added mass. Doing so, the parametric analysis is performed with only one calculation and the curve of LP
in Fig. 2b provides the amplitude of the central beam after the symmetry-breaking induced by an added mass.
22.15 22.2 22.25
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Ω
W
m
ax
5×10−4
3×10−4
2×10−4
10−4
8×10−5
5×10−5
10−5
10−12, 10−8, 10−6
(a) Responses curves for multiple added masses
22.15 22.2 22.25
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Ω
W
m
ax
5×10−4
1
3
3
2
(b) Limit-point tracking procedure. 1: Response Curve,
2: Limit Point Localisation, 3: Limit-point tracking
Figure 2: Response curves of the central resonator for a varying non-dimensional added mass.
Conclusions
A parametric analysis of a three-resonator array has been performed, based on a symmetry-breaking event in order to
detect and measure an added mass. The parametric analysis based on the direct continuation of Limits Points constitutes
a very efficient method for the numerical estimation of the measured mass with only one calculation. This research
represents a step towards the implementation of MEMS-based mass spectrometry using arrays of resonators.
References
[1] S. Gutschmidt and O. Gottlieb. Nonlinear dynamic behavior of a microbeam array subject to parametric actuation at low, medium and large dc-
voltages. Nonlinear Dynamics 67(1):1–36, 2012.
[2] V-N Nguyen, S. Baguet, C-H Lamarque, R. Dufour: Bifurcation-based micro-/nanoelectromechanical mass detection. Nonlinear Dynamics
79(1):647–662, 2014.
[3] N. Kacem, S. Baguet, S. Hentz, and R. Dufour: Computational and quasi-analytical models for non-linear vibrations of resonant MEMS and NEMS
sensors. International Journal of Non-Linear Mechanics 46(3):532–542, 2011.
[4] L. Xie, S. Baguet, B. Prabel and R. Dufour: Bifurcation tracking by Harmonic Balance Method for performance tuning of nonlinear dynamical
systems. Mechanical Systems and Signal Processing, In press, 2016. doi:10.1016/j.ymssp.2016.09.037.


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Mass detection through parametric analysis and symmetry-breaking in a MEMS array

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