1. Introduction
Atomic force microscopy (AFM) is a scanning probe microscopy (SPM) technique to obtain images of surface topography at the atomic scale, in a noninvasive manner, from a wide variety of samples on a scale from angstroms to 100 microns [1]. A typical AFM consists of a microcantilever with a sharp tip, a sample positioning system, a detection system, and a control system. The associated length scales are sufficiently small to call the applicability of classical continuum models into question [2]. In this work, we seek to develop a vibration dynamics framework for beams that include smart materials and subject to surface effects. This framework is also considered in the case of a two-span beam in which the first segment is governed by the Timoshenko model and the second segment is an Euler-Bernoulli beam model [3].
Recently, new generations of active microcantilevers have included piezoelectric materials locally attached at the microbeam with the role of sensors and/or actuators linear and nonlinear [4–9], among others. This has led to the study of multispan beams for AFM. The inclusion of smart materials layers will modify material properties between neighboring segments, producing discontinuities and fulfilling compatibility conditions for the continuity of the displacement and rotation of the beam and for the equilibrium of bending moment and shear at the discontinuity points.
Surface effects often play a significant role in the physical properties of micro- and nanosized materials and structures. Since the atoms within a very thin layer near surfaces experience a different local environment form that is experienced by atoms in bulk, the physical properties and mechanical response of surfaces will be distinct from those of bulk materials. Contrary to macroscopic structures, surface effects can strongly influence the stress and deformation properties of nanodevices. This latter is due to the increasing ratio between surface/interface area and volume. For instance, for a bulk deformation energy
In some models, considering effective properties (
Simulations have been performed for triple span beams with and without a patch in the first segment and with loads in the first and last segment with tip-sample interaction. These loads included a uniform pulse load and its second derivative associated with the moment due to a piezoelectric patch. It is observed how the inclusion of piezoelectric materials absorbs vibrations when compared with classical multispan beams. Also, the robustness of the impulse response method with varying parameters and its influence in the behavior of the responses are observed.
The size dependence effects in Timoshenko microbeams with surface effects (TMB) have been simulated for the nondimensional natural frequency and compared with those of the classical Timoshenko beam model (TB). It has been observed that for beam length on the order of nanometer to microns, the difference between natural frequencies is apparent, and by increasing the length of the microbeam, the results tend to Timoshenko classical theory; that is, the surface effects are significant only in nanoscale. This same behavior was observed in [14] for a microbeam simply supported.
This work involves a methodology that may be adequately adapted to study recent advances in micro-/nanosized structures that are intensively used to design advanced micro-/nanosensors and molecular transportation systems devices for various engineering and medical applications. These sensors due to their ultrahigh resonant frequencies are important in sensitive sensing, molecular transportation, molecular separation, high-frequency signal processing, and biological imaging. Some deposited processes in the formation of thin-film composite membranes can be similar to types of multispan vibrating beams. However, the complex configuration of materials used in real world devices requires the study of material properties stemming from their atomically thin layered structures. Chemical deposited processes in the manufacturing of thin films on semiconductors can involve low temperatures with silicon rich nitride. Conventional structural analysis methods assume ideal structures (free of irregularities) but material and/or geometrical variations in a structure may result in drastic changes in its dynamical behaviors. Diffusion- and reaction-controlled interfacial polymerization is an important and practical topic that is beyond our scope [15–19].
This paper is organized as follows. In Section 2, we consider the Euler-Bernoulli multispan beam model, their compatibility conditions, and the boundary conditions in a matrix form. In Section 3, the dynamic response of the matrix Euler-Bernoulli multispan beam model subject to tip-sample interactions and external forcing is given in terms of the distributed matrix impulse response. The case of a microcantilever with a piezoelectric layer is discussed in Section 4. In Section 5 are presented the results of a double-span beam, obtained by the expansion of the first segment of the triple piezoelectric beam. In Sections 6 and 7 is considered the eigenanalysis of Timoshenko beam models with surface effects and their comparison with the Timoshenko classical model, for micro- and nanoscale. In Section 8, we consider the multispan Timoshenko beam model. AFM-based nanoscale processing with continuum surrounding media such as that found in biology and nanomachining applications [20–22] suggests observing frequency effects that arise with an academic two span beam model in which one segment includes rotatory inertia and shear deformation and the other one neglects both effects.
2. Flexural Vibrations Using the Euler-Bernoulli Multispan Beam Model
We consider a multispan microcantilever of length
2.1. Matrix Formulation
The stepped Euler-Bernoulli model (1) can be written as a second-order differential equation
where
2.2. Boundary and Internal Conditions for a Multispan Cantilever Beam
For a cantilever beam, the boundary conditions can be written in a compact matrix form
At the points
3. Forced Vibrations
The dynamic response of a forced multispan Euler-Bernoulli model
To work on the frequency domain, we need to introduce transfer function
It turns out that
where
The procedure mentioned above is also related to the Riemann function method for integrating partial differential equations. Dynamic responses have been considered in the field of control of distributed systems and in elastodynamics in connection with vibrations and cracking problems [23–26].
In practice, when computing the convolution integral, which corresponds to the forced response
3.1. Frequency Response
Harmonic and piecewise linear forcing are of interest in frequency analysis. When seeking a response of the same type, the transfer function is introduced. Given the harmonic input
In particular, for a concentrated force at a point
As before, by substituting the initial values in (14), the induced free response will now be
3.2. Closed Form of the Transfer Function for a Multispan Cantilever Beam
The Green function of a multispan Euler-Bernoulli can be determined by using an appropriate solution basis in each segment. From (12),
The above process will be carried out for the case of a two stepped cantilever beam with the intermediate discontinuity point
The boundary conditions and the initial values of
Thus, for
By applying the compatibility conditions
we get the algebraic system
Remark. The case of axially loaded stepped beams has been investigated in [28] with nonclassical conditions. There, the Green function was obtained by working with a solution basis that considers hyperbolic and trigonometric functions in the general solution. These functions depend upon simple roots of a complete quadratic polynomial due to the inclusion of a second derivative spatial term in (24). When the axial load is removed, the solution basis reduces to the one considered for particular cases of vibrating beams. We observe that through a limit process the use of
4. A Cantilever Beam with a Piezoelectric Layer in AFM
Active microcantilever beams due to their structural flexibility and sensitivity to atomic and molecular forces have received increased attention in a variety of nanoscale sensing and measuring applications, including atomic force microscopy, thermal scanning microscopy, and biomass sensing.
In [9], a microcantilever model was proposed for studying atomic force microscope. It was formulated as a three stepped beam with a piezoelectric layer patch in the first segment and the other two segments were simple beams with different cross-sectional areas according to Figure 1(a).
[figures omitted; refer to PDF]
The governing equations can be written as a system of three second-order partial differential equations with constant coefficients
The parameters
The values of these parameters [9] are presented in Table 1.
Table 1
Parameters of piezoelectric multispan beam.
Parameter | Numeric Value | Unity |
---|---|---|
Length | | m |
Length | | m |
Length | | m |
Width | | m |
Width | | m |
Width | | m |
Thickness | | m |
Thickness | | m |
Density | 2330 | |
Density | 6390 | |
Young’s modulus | 105 | |
Young’s modulus | 104 | |
The distributed cross-sectional moment
4.1. Forced Responses
For homogeneous boundary conditions and null initial data, we have from (13) the forced response
The modal approximation
has been used in simulations with and without the inclusion of a piezoelectric layer. Expansions were truncated with a small number
In Table 2 are showed the first six natural frequencies obtained in this work for a triple span beam with piezoelectric patch and without it. The first three frequencies are compared with those obtained experimentally in [9]. There is an agreement with the methodology of this work and it can be observed that the elimination of piezoelectric layers diminishes the frequency values. In Figure 2 are shown the corresponding vibration modes; in the first segment the modes of piezoelectric beam have smaller amplitude.
Table 2
Comparative natural frequencies between cases of double and triple span beams with and without piezoelectric patch on the first segment.
Case | Piezoelectric path | Natural frequencies | |||||
---|---|---|---|---|---|---|---|
| | | | | | ||
Triple span [9] | with piezoelectric path | 52.3 | 203.0 | 382.5 | - | - | - |
Triple span (Figure 1(a)) | with piezoelectric path | 50.6 | 209.2 | 393.7 | 910.7 | 1412.5 | 2016.7 |
Double span (Figure 1(b)) | with piezoelectric path | 46.2 | 222.9 | 376.2 | 875.3 | 1537.5 | 1948.6 |
Double/Triple span | without piezoelectric path | 27.3 | 142.9 | 320.1 | 618.9 | 1073.0 | 1579.4 |
| |||||||
|
The effects of the inclusion of a piezoelectric patch are appreciated by applying a harmonic rectangular pulse load in the first segment and in the third segment for a three stepped beam with and without piezoelectric layer. It is observed that the spatial amplitude diminishes with the inclusion of a piezoelectric patch when the pulse is positioned in the piezoelectric layer segment; this effect does not appear when the pulse is positioned in the third segment (Figure 3).
[figure omitted; refer to PDF]In Figure 4 are compared the forced responses due to forcing terms only in the first segment. The first output tracks the forcing input
5. Double-Span Microbeam with Material and Geometric Discontinuities
We consider a double-span beam of length
The first six natural frequencies of this beam, with and without piezoelectric patch, were given in the third and fourth rows of Table 2 and were obtained by the modal method. We observed that the reduction of three segments for two segments, maintaining the others material and geometric properties, influences the magnitude of the natural frequencies. When the piezoelectric patch is removed, then the frequencies also decrease. The results were the same as those of the case of three stepped beam when removing the piezoelectric patch.
In Figure 5, by using the methodology described from (24) to (35), is presented a 3D graphic of the Green function
6. Surface Elasticity and Residual Surface Tension in the Timoshenko Beam Model
The Timoshenko beam model has been modified with the inclusion of the residual tension and surface elastic modulus [14]. By using the energy method with the elastic strain energy of surface induced in the potential energy and residual surface tensions, expressed by Laplace-Young equations, it was obtained the modified Timoskenko model
6.1. Matrix Formulation
The coupled Timoshenko model (48) can be written as a second-order differential equation with matrix coefficients
6.2. Eigenanalysis
The search of exponential solutions
By using the initial values of h(x) in (57) and the clamped boundary condition
We thus have the reduced system
The fundamental matrix response
We observe from Figure 6 that for frequencies not above the critical frequency, the fundamental response
For a microcantilever beam described by the Timoshenko model with surface effects, the size dependence in the natural frequency of Timoshenko classical model and Timoshenko model including surface effects is illustrated in Figure 7. The solutions based on classical Timoshenko beam theory and Timoshenko beam theory including surface effects are denoted by TB and TMB, respectively. The fundamental natural frequencies are normalized to fundamental frequency of cantilever Euler-Bernoulli beam. In this figure are considered the parameters utilized in [14] for the same purposes. The parameters of surface elasticity and residual surface tension can be determined by molecular dynamics simulations or experiments. Residual surface stresses can be either positive or negative, depending on the crystallographic structure [33]. For an anodic alumina
In Figure 7, the size dependence effects in the nondimensional natural frequency of TMB microbeams in comparison to the classical TB are illustrated. We can observe that for beam length on the order of nanometer to microns, the difference between natural frequencies is apparent and, by increasing the length of the microbeam, the results tend to Timoshenko classical theory; that is, the surface effects are significant only in nanoscale. This same behavior was observed in [14] for a microbeam simply supported.
7. Timoshenko Model with Surface Energy in Thick Nanobeams
In [13], the surface effects are included in the Timoshenko beam model following Gurtin-Murdoch continuum theory [34], in which is considered an elastic surface with zero thickness fully bonded to its bulk material. This elastic surface adds a set of specific constitutive equations relative to distinct material properties of the surface and surface energy effects. Let us denote an isotropic material beam, with rectangular transversal section, length
In a compact notation we have
where
and
The material parameters of bulk are
Rewriting (74), we have
Considering exponential solutions in the form
where
In order to guarantee
7.1. Singular Eigenvalue Problem
By assuming exponential solutions
The case of simple roots can be characterized from the condition
Lemma 1.
For the case of distinct roots
The above representation of
7.1.1. Repeated Roots Analysis
From the given characterization of simple roots, it is clear that the repeated roots can be only
Lemma 2.
From the above lemmas, critical frequencies values will arise for natural frequencies
(i)
(ii)
7.2. Frequency Equations
Frequency equations will arise when applying boundary conditions as given below.
7.2.1. Simply Supported Beam
In this case we have the following boundary conditions, at
From the boundary conditions at
Since
7.2.2. Cantilever Beam
In this case the boundary conditions are given by
In Table 3 and Figure 8, we present numerical results about natural frequencies and mode shapes. Both cases of supported-supported (S-S) and cantilever (C) boundary conditions are considered. The simulation was done with a silicon beam having the following material and surface parameters [13]:
Table 3
Comparison between natural frequencies of Timoshenko beam models including and not including surface effects, considering nano- and microscale, of a Silicon beam. Nanobeam geometric parameters:
Nano | Micro | ||||||||
---|---|---|---|---|---|---|---|---|---|
BC | Model | Natural frequencies | Natural frequencies | ||||||
| | | | | | | | ||
S-S | TMB | 7.0808 | 25.063 | 51.327 | 82.922 | 7.1996 | 27.010 | 55.669 | 89.756 |
TB | 7.1998 | 27.012 | 55.673 | 89.763 | 7.1998 | 27.012 | 55.673 | 89.763 | |
TMB [13] | 7.08 | 25.07 | 51.33 | 82.92 | - | - | - | - | |
TB [13] | 7.20 | 27.02 | 55.70 | 89.76 | - | - | - | - | |
| |||||||||
C | TMB | 3.0168 | 14.674 | 36.484 | 64.514 | 2.5976 | 15.273 | 39.297 | 69.725 |
TB | 2.597 | 15.273 | 39.300 | 69.730 | 2.5971 | 15.273 | 39.300 | 69.730 | |
TMB [13] | 3.02 | 14.67 | 36.38 | 64.46 | - | - | - | - | |
TB [13] | 2.60 | 15.28 | 39.31 | 69.74 | - | - | - | - |
Comments
(i) From Table 3 and Figure 8, it is observed that the results agree with those found in the literature [13, 14].
(ii) When the beam length increases, from nano- to microscale, the results tend to the classical results.
(iii) Natural frequencies can increase or decrease compared to classical results, depending on elastic surface constants signs and boundary conditions (cantilever beam is more influenceable than simply supported beam).
(iv) According to Figure 8, the mode shape component
8. Multispan Timoshenko Beams
The same formulation, which was presented in Section 2 for Euler-Bernoulli multispan beams, can be used with the Timoshenko multispan beam model. We replace the scalar components that appear in the block matrices defined after (1), by the
General boundary conditions can be written as
The continuity conditions for displacement, gyro, flexural moment, and shear force in the transversal section discontinuity points
In what follows, we shall consider double-span beams in which the first segment is modeled by the Timoshenko beam model and the second segment can be modeled by a Timoshenko (TB) or an Euler-Bernoulli (EB) model.
TB-TB. In this case,
For a simply supported beam we have
The compatibility conditions, at the point
The spatial amplitude of a free transversal vibration, at each segment,
The boundary conditions, obtained from (109), can be conveniently written as
We choose the fundamental basis at each segment, considering convenient translations,
The matrix fundamental solution
Then
Replacing (128) in the boundary and compatibility conditions we obtained a system
For a cantilever beam we have
The matrix
where
TB-EB. We now consider that the movement in the second segment is governed by the Euler-Bernoulli beam model (TB-EB). For simplicity, we shall consider Euler-Bernoulli compatibility conditions at the discontinuity point.
In this case, for
The problem can be also described in matrix form as
The general boundary conditions are given in
In the discontinuity point
where
The boundary conditions can be written as
The solutions
8.1. Simulations
The two span beam models considered have been compared with results of Euler-Bernoulli and Timoshenko double-span beam models in dimensional and unidimensional formulations.
Dimensional Model. We assume a double-span beam, with transversal section discontinuity due to variable thickness. This beam has constant width, but different thickness. The geometric and materials properties of this beam are presented in Table 4.
Table 4
Parameters of the double-span beam [29].
Parameters | Numeric value | Unity |
---|---|---|
Length | 0,254 | m |
Length | 0,140 | m |
Width | 0,02545 | m |
Thickness | 0,01905 | m |
Thickness | 0,00549 | m |
Mass density | 2830 | |
Young’s modulus | 71,7 | |
Moment of inertia | | |
Moment of inertia | | |
We present results about natural frequencies, of three cases: two Euler-Bernoulli segments (EB-EB), two Timoshenko segments (TB-TB), and coupled case (TB-EB), in Table 5.
Table 5
Comparison of natural frequencies from the double-span beam. EB-EB: two segments of Euler-Bernoulli beam, TB-TB: two segments of Timoshenko beam, and TB-EB: first segment using Euler-Bernoulli beam model and second segment using Timoshenko beam model.
Case | Model | Natural frequencies | |||
---|---|---|---|---|---|
| | | | ||
S-S | EB-EB | 101.77 | 782.09 | 1300.56 | 2874.69 |
TB-EB | 101.74 | 777.85 | 1291.65 | 2816.98 | |
TB-TB | 101.64 | 775.85 | 1285.65 | 2796.89 | |
| |||||
C-F | EB-EB | 139.56 | 326.69 | 1170.27 | 1797.41 |
TB-EB | 139.39 | 325.84 | 1154.90 | 1771.78 | |
TB-TB | 139.31 | 325.45 | 1150.82 | 1761.06 | |
| |||||
F-F | EB-EB | 292.42 | 1181.28 | 1804.01 | 3603.04 |
TB-EB | 292.14 | 1775.94 | 1775.8 | 3512.95 | |
TB-TB | 291.77 | 1167.89 | 1786.63 | 3483.40 | |
[Ref] | 286-291 | 1159-1165 | 1759-1771 | - |
We observe from Table 5 that the values of natural frequencies of the coupled beam (TB-EB) are between the respective values obtained by TB-TB and EB-EB cases, wherein the EB-EB natural frequencies, as expected, are greater than the others, in all boundary conditions cases presented: S-S (simply supported), C-F (clamped free), and F-F (free-free). Moreover, in the case of free-free boundary condition, as expected, TB-TB produces the most similar results to those obtained experimentally in [29].
Dimensionless Model. We now consider the two span models with classical dimensionless variables. For the Euler-Bernoulli beam model, the spatial problem in a dimensionless form is given by
where the dimensionless parameters are
The Timoshenko model in a dimensionless matrix form is given by
with the dimensionless parameters defined by
Then we have the dimensionless frequencies by each segment
These frequencies can be written as a relation of
In Table 6 we present results about dimensionless natural frequencies; we have considered variations in the position of the discontinuity point
Table 6
The effects of the change of discontinuity point
Configuration A | Configuration B | ||||||||
---|---|---|---|---|---|---|---|---|---|
Case | Model | Dimensionless natural frequencies | |||||||
| | | | | | | | ||
| TB-TB | 1.4093 | 3.4619 | 5.6243 | 7.4545 | 1.7184 | 4.0974 | 6.5148 | 8.7633 |
TB-EB | 1.4098 | 3.4721 | 5.6637 | 7.5438 | 1.7193 | 4.1198 | 6.5990 | 8.9637 | |
EB-EB | 1.4101 | 3.4769 | 5.6854 | 7.6159 | 1.7209 | 4.1381 | 6.6554 | 9.077 | |
| |||||||||
| TB-TB | 1.9972 | 3.5642 | 5.9015 | 8.2377 | 2.0150 | 4.0913 | 6.8545 | 9.1581 |
TB-EB | 1.9985 | 3.5725 | 5.9372 | 8.3063 | 2.0162 | 4.1061 | 6.9121 | 9.3086 | |
EB-EB | 2.0004 | 3.5901 | 5.9831 | 8.5546 | 2.0198 | 4.1331 | 7.050 | 9.5862 | |
| |||||||||
| TB-TB | 2.0982 | 4.7587 | 6.6143 | 9.1855 | 1.9918 | 4.7325 | 7.3808 | 9.7778 |
TB-EB | 2.1000 | 4.7673 | 6.6436 | 9.2371 | 1.9933 | 4.7395 | 7.4083 | 9.8619 | |
EB-EB | 2.1046 | 4.8267 | 6.7614 | 9.6670 | 1.9973 | 4.8113 | 7.6241 | 10.329 |
Configuration A: double-span beam with variable thickness and constant width,
Configuration B: double-span beam with variable thickness and constant width,
Comparative results of dimensionless natural frequencies are presented in Table 6.
We observe from Table 6 that the values of dimensionless natural frequencies of the coupled beam (TB-EB) are between the respective values obtained by TB-TB and EB-EB cases, wherein the EB-EB natural frequencies, as expected, are greater than the others. Also, we can observe that when the thickness of the second segment increases, the natural frequencies become greater, mainly the higher frequencies (third and fourth frequencies), except for the case
9. Conclusions
This paper formulated in matrix form the time domain determination of forced responses and in the frequency domain the obtention of modes of elastic models that can be used in AFM when subjected to tip-sample interactions.
Forced responses are determined by convolution of the input load with the time domain impulse matrix function. The corresponding matrix transfer function and modes of a multispan cantilever beam were determined in terms of solution basis which have the same shape and are generated by a fundamental solution.
It was observed that the spatial amplitude diminishes with the inclusion of a piezoelectric patch when the pulse is positioned in the piezoelectric layer segment; this effect does not appear when the pulse is positioned in the third segment. The reduction of three segments for two segments, maintaining the others material and geometric properties, influences the magnitude of the natural frequencies. When the piezoelectric patch is removed, then the frequencies decrease.
We worked with two Timoshenko beam models including surface effects through different approaches and changing the classical Timoshenko equations in different manners. For both models were presented the results which evidence that, in agreement with the literature [13, 14], for beam length on the order of nanometer to microns, the difference between natural frequencies is apparent and, by increasing the length of the microbeam, the results tend to Timoshenko classical theory; that is, the surface effects are significant only in nanoscale.
Simulations were performed with double-span beams in which the first segment is modeled by the Timoshenko beam model and the second segment can be modeled by a Timoshenko (TB) or an Euler-Bernoulli (EB) model. Consistent results of TB-EB beams were obtained, in dimensional and dimensionless forms, when compared with EB-EB and TB-TB cases. The value of frequencies for this model (TB-EB) is located between the natural frequencies of uniform double-span models TB-TB and EB-EB.
Conflicts of Interest
We have no conflicts of interest.
Acknowledgments
This work was performed as part of the employment of the authors at their institutions and the third author held a fellowship no. 140823/2014-0 from CNPq of Brazil.
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Abstract
Vibration dynamics of elastic beams that are used in nanotechnology, such as atomic force microscope modeling and carbon nanotubes, are considered in terms of a fundamental response within a matrix framework. The modeling equations with piezoelectric and surface scale effects are written as a matrix differential equation subject to tip-sample general boundary conditions and to compatibility conditions for the case of multispan beams. We considered a quadratic and a cubic eigenvalue problem related to the inclusion of smart materials and surface effects. Simulations were performed for a two stepped beam with a piezoelectric patch subject to pulse forcing terms. Results with Timoshenko models that include surface effects are presented for micro- and nanoscale. It was observed that the effects are significant just in nanoscale. We also simulate the frequency effects of a double-span beam in which one segment includes rotatory inertia and shear deformation and the other one neglects both phenomena. The proposed analytical methodology can be useful in the design of micro- and nanoresonator structures that involve deformable flexural models for detecting and imaging of physical and biochemical quantities.
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Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer
Details

1 PGMAT, Departamento de Matemática, CCNE, Universidade Federal de Santa Maria, Av. Roraima 1000, 97105-900, Santa Maria, RS, Brazil; PPGMAp-IME, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves, 9500, 91509-900, Porto Alegre, RS, Brazil
2 PGMAT, Departamento de Matemática, CCNE, Universidade Federal de Santa Maria, Av. Roraima 1000, 97105-900, Santa Maria, RS, Brazil
3 Centro de Engenharias, CENG, Universidade Federal de Pelotas, Almirante Barroso, 1734, Pelotas, RS, 96010-280, Brazil
4 PPGMAp-IME, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves, 9500, 91509-900, Porto Alegre, RS, Brazil