Online proceedings - EDA Publishing Association

11-13
May 2011, Aix-en-Provence, France
Dynamic Behavior of Resonant Piezoelectric
Cantilevers Partially Immersed in Liquid
M. Maroufi 1,2 ,Sh. Zihajehzadeh 1,3 , M. Shamshirsaz 1 , A.H. Rezaie 3 , M.B. Asgari 4
1 New Technologies Research Center, 2 Mechanical Engineering Department, 3 Electrical Engineering Department
Amirkabir University of Technology (Tehran Polytechnic), 4 Niroo Research Institute
424 Hafez Ave., P.B. 15875-4413. Tehran, Iran
E-mail: shamshir@aut.ac.ir
Abstract
Resonant Piezoelectric-excited Millimeter-sized Cantilevers
(PEMC), has attracted many researchers' interest in the
applications such as liquid level and density sensing. As in
these applications, the PEMC are partially immersed in liquid,
an appropriate analytical model is needed to predict the
dynamic behavior of these devices.
In this work, a PEMC has been fabricated for liquid level
sensing. An analytical model based on Euler-Bernoulli theory
and energy method is developed and applied to evaluate the
performance of this device with respect to different tip
immersion depth. To validate this model, the theoretical
results are compared with the experimental results for the tip
immersion depth from 0.5 mm to 9 mm in water. The
simulation results are in almost good agreement with
experimental data. The difference in natural frequency
obtained by the theoretical model for different immersion
depth remains less than 8%. The linear region of the natural
frequency shift versus immersion depth has been identified to
be from the depth of 9 to 11 mm.
I. INTRODUCTION
Nowadays, resonant Piezoelectric-excited Millimeter-sized
Cantilevers (PEMC) have many applications as sensors.
Among these diverse applications, are the ones where the
cantilever is partially immersed in the liquid environment. In
these cases, PEMC are used for online measuring of liquid
density [1], [2], [3] or online determination of liquid level at
micron resolution [4]. Online level detection of liquid is a
powerful tool in many analytical processes where solvent
concentration has to be monitored.
Even though, there exists different tools for liquid level
sensing such as ultrasonic, acoustic and optical methods, none
of them is competent with PEMC, considering their ease of
fabrication, small size and high performance [4].In fact, the
performance of these devices for sensing application in liquid
environment depends on many factors such as dimension of
the cantilever and the piezoelectric layer, the immersion depth
of the cantilever into liquid and so on.
To evaluate the performance of PEMC partially immersed in
liquid, a theoretical model is needed. Analytical model for the
piezoelectric driven macro cantilever in air in introduced in [5]
and also a model for the thermal driven cantilever wholly
immersed in liquid with application in AFM is presented in
[6].
In this work, a PEMC has been fabricated for liquid level
sensing. The motivation is first to develop an analytical model
to predict the dynamic behavior of PEMC partially immersed
in liquid. This model is derived here based on Euler-Bernoulli
theory and energy method. Further, this model could be
utilized to investigate the effect of the different geometrical
and material properties on the performance of these devices as
future work.
Second objective in this work is to identify the appropriate
immersion depth range in which the resonant frequency
changes due to immersion depth variation show a linear
behavior in liquid level sensor application.
To validate this model, the theoretical results are compared
with the experimental results for the tip immersion depth from
0.5 mm to 9 mm in water. The simulation results are in almost
good agreement with experimental data. The difference in
natural frequency obtained by the theoretical model for
different immersion depth remains less than 8%. The linear
region of the natural frequency shift versus immersion depth
has been identified to be from the depth of 9 to 11 mm.
II. THEORETICAL MODEL
The fabricated PEMC is depicted schematically in Fig. 1. This
structure consists of a millimeter sized steel beam as a
cantilever on which a piezoelectric patch is attached. The
cantilever is immersed partially in the fluid. Applying
electrical AC voltage on the piezoelectric patch, the cantilever
is forced to vibrate.
To model the resonant cantilever partially immersed in liquid ,
three regions on the cantilever has been considered; a) first
part where piezoelectric patch is bonded on the cantilever, b)
middle part of cantilever where it vibrates freely ignoring air
damping effect c) end part where the cantilever vibrates in the
liquid. Also, three coordinate systems are assumed in each
region (Fig. 1).
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