=Paper=
{{Paper
|id=Vol-2485/paper21
|storemode=property
|title=Visualization of Contact Interaction of Nanobeams
|pdfUrl=https://ceur-ws.org/Vol-2485/paper21.pdf
|volume=Vol-2485
|authors=Maxim Zhigalov,Victor Apryskin,Vadim Krysko
}}
==Visualization of Contact Interaction of Nanobeams==
https://ceur-ws.org/Vol-2485/paper21.pdf
Visualization of Contact Interaction of Nanobeams
M.V. Zhigalov1, V.A. Apryskin1, V.A. Krysko1
zhigalovm@yandex.ru|wwooow@yandex.ru
1
Yuri Gagarin Saratov State Technical University, Saratov, Russia
The paper presents a visualization of the contact interaction of two Bernoulli-Euler nanobeams connected through boundary
conditions. Mathematical models of beams are based on the gradient deformation theory and the theory of contact interaction of
B. Y. Cantor. The visualization is based on Fourier transform and wavelet transform, phase portrait.
Keywords: Bernoulli-Euler nanobeam, gradient deformation theory, contact problem, Fourier transform, wavelet.
gradient vector, the deviator tension gradient tensor, and the
1. Introduction symmetric rotation gradient tensor.
In this paper, a mathematical model of Bernoulli-Euler
The designs of modern devices are complex multi-layer
nanobeams connected through boundary conditions under the
packages with small gaps between the elements, so an
action of transverse load is constructed. Three material length
important issue is to take into account the contact interaction of
scale parameters are introduced to account for dimensional
layers, which in turn leads to a strong nonlinearity -
effects (l0 , l1, l2 ) . To account for the contact between the
constructive. Such nano electro-mechanical systems (NEMS)
are widely used in various electronic devices, in particular in beams, a Winkler coupling between the compression and the
gyroscopes (layered flat micromechanical accelerometers contact pressure between the two beams is used [13]:
(MMA). Note that the account of contact interaction leads to the 1
1 sign w1 hk w2 , (1)
chaotic state of the system already at small amplitudes of 2
oscillations. The presence of a gap between the elements, such where 1, if w1 w2 hk то that is, there is contact
as beams, already at small deflections, commensurate with the
gap between the elements, can lead the object under study in a between the plate and the beam, else 0 , w1, w2 , hk -
state of chaotic oscillations. Therefore, the need for research deflections of the first and second beams and the gap between
with visualization of the contact interaction of nano-mechanical them, respectively.
systems in the form of beams is undoubtedly relevant and The mathematical model of contact between two
requires attention. nanobeams, based on the kinematic Bernoulli-Euler hypothesis,
is described by a system of resolving equations:
2. Statement of the problem 4 wm bh 3 96 2
2 l02 bh l22 bh l1 bh
Micro-and nano-sized beams are widely used in various x 4
12 225
micro-and nano-Electromechanical systems (vibration sensors
6 wm 2 bh3 14 2 bh3
[1], micro-drives [2], micro-switches [3]). The dependence of 2l0 l1 qm (2)
elastic behavior on the body size at the micron scale was x 6 12 225 12
observed experimentally in different substances (metals [4, 5] 2w w
and alloys [6], polymers [7], crystals [8]). (1) m K ( w1 w2 h ) ,
t 2 t
The dependences of elastic behavior on size can be
explained using molecular dynamics (MD) simulations or where m – beam number (m=1,2), h - the gap between the
higher order continuum mechanics. Although the molecular beams.
dynamics approach can provide more accurate approximations Boundary conditions are:
to real objects, it is too expensive from a computational point of wm
view. Therefore, the higher order continuum mechanics wm 0; 0. (3)
approach has been widely used in the modeling of small-scale x
structures. Initial conditions are:
The development of a higher order continuum theory can be w x,0
wm ( x,0) 0; m 0. (4)
traced back to the earliest work of Piola in the 19th century, as t
shown in [9], and the work of the Kosser brothers [10] in 1909. The system (2-4) was reduced to the Cauchy problem using the
However, the ideas of the Kosser brothers received considerable finite difference method O(h2). The Cauchy problem was solved
attention from researchers only since the 1960s, when a large by the Runge-Kutta method of 4 orders. A study of convergence
number of higher-order continuum theories were developed. In by the method of finite differences, on the basis of which the
general, these theories can be divided into three different optimal number of partitions was chosen, was carried out. The
classes, namely, the family of strain gradient theories, the partitioning step for the Runge-Kutta method was determined
microcontinuum, and nonlocal elasticity theories. according to the Runge principle.
Based on the higher order stress theory of Mindlin [11] and
Lam et al. [12] proposed the theory of elasticity of the 3. The results of the study of the influence of
deformation gradient, in which, in addition to the classical length scale factors on the nature of oscillations
equations of equilibrium of forces and moments, a new
additional equilibrium equation is introduced, which determines For the considered tasks the following parameter values
the behavior of stresses of higher orders and the equation of were used: a / h 30 , q1 10sin 5.3 t , q2 0 , 0.1,
equilibrium of moments. Three parameters of the material
length scale are introduced for isotropic linear elastic materials h 0.01.
(l0 , l1, l2 ) . According to this theory, the total strain energy The research results for two of the nine considered
density is a function of the symmetric strain tensor, the dilation combinations of coefficients (l0 , l1, l2 ) are shown in Figure 1,
2. For the first case (Fig. 1), all coefficients are zero, i.e.
considered beams on the classical theory. The second case (Fig.
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
2) is characterized by the values l0=0.3; l1=0.3; l2=0.3, i.e. all The Fourier spectrum for both beams indicates a chaotic
three dimensional factors are taken into account. The figures in state of the system, while both spectra are qualitatively similar
the first row show the deflections of the first w1 and second and have several pronounced frequencies in the low range
beams w2 . The second line shows the phase synchronization of i 3.5;1.5 . Wavelet spectra indicate that these frequencies
oscillations 1 2 , marked in dark color. The third and fourth appear in the spectrum with t 200 . Moreover, the frequency
spectrum varies significantly over time. Phase portraits of beam
lines show the Fourier spectrum S ( ) , the wavelet spectrum
oscillations are “mirrored”.
based on the Morlet maternal wavelet, and the phase portrait
w w for the first and second beams, respectively. Phase
synchronization is based on the approach described in [14-15].
When analyzing phase chaotic synchronization, it is necessary
to enter the phase (t ) of chaotic signals [16, 17]. Frequencies
of chaotic signals are defined as average rates of change of the 2a) deflections of the first w1 and second w2 beams
phases of signals (t ) . Wavelet analysis can be applied to
identify the phase synchronization mode of mechanical
dynamic systems with a poorly defined phase. The behavior of
such systems is characterized on the basis of continuous sets of
phases, which are determined using continuous wavelet 2b) phase synchronization
transform of a chaotic signal w(t ) [16, 17]. If the beams in
question are not in the phase synchronization mode, their
behavior is asynchronous on all time scales. As soon as any of
the time scales of the considered dynamic systems are
synchronized, the phase synchronization mode arises.
Obviously, the time scales that have the maximum power of the 2c) Fourier 2d) Wavelet 2e) Phase portrait
wavelet spectrum are synchronized first of all. The remaining spectrum
time scales are still unsynchronized. Therefore, phase
synchronization leads to the occurrence of phase capture on
synchronized time scales.
2f) Fourier 2g) Wavelet 2e) Phase portrait
spectrum
Fig. 2. Results for the case l0=0.3; l1=0.3; l2=0.3.
Consider the case when both beams are described according
1a) deflections of the first w1 and second w2 beams to the theory of the strain gradient taking into account the
values of l0=0.3; l1=0.3; l2=0.3. Graphs of deflections (signals)
differ significantly from the previous case. The synchronism of
oscillations is present at a longer interval up to t 200 . The
phase synchronization graph shows that in this time interval
synchronization occurs at two frequencies: at the excitation
1b) phase synchronization frequency p 5.3 and at the frequency 2 , as in the
previous case. However, synchronization at the excitation
frequency is uniform over the entire time interval, in contrast to
the previous case, where phase synchronization is practically
absent in the interval t 480; 600 . The Fourier spectrum for
1c) Fourier 1d) Wavelet 1e) Phase portrait both beams is qualitatively similar, as in the previous case, but
spectrum there are differences. The number of frequencies with the
highest energy (peaks in the spectrum) is much larger and their
distribution differs significantly from the previous case. Thus,
the contact interaction energy of two nanobeams differs
significantly from the beams according to the classical theory,
1f) Fourier and therefore, when studying the contact of NEMS elements, it
1g) Wavelet 1h) Phase portrait is necessary to use the theory of size-dependent behavior. The
spectrum
wavelet spectrum also has a significant difference from the
Fig. 1. Results for the case l0=0; l1=0; l2=0.
previous case. For the first beam, both the excitation frequency
Graphs deflection changes in time is shown on the complex
and all frequencies of the range are present throughout the time
vibrations. And before the t 160 deflections change
interval 0;3 . The frequencies of the other bands appear at
synchronously. This reflects the phase synchronization graph in
the second row of the table. There are two synchronization the initial time t 0; 230 and at the end of the time interval
zones on this graph: at the excitation frequency p 5.3 and at
for contact interaction studies. The phase portraits of both
the lower frequency 2 . In the rest of the time interval, the beams, as well as for the previous case, are “mirrored”, but
oscillations of the two beams are practically “mirrored”, and qualitatively different. They also differ quantitatively by the
synchronization occurs only at the excitation frequency values of the velocity of oscillations w .
p 5.3.
4. Conclusion [16] Osipov G.V., Pikovsky A.S., Rosenblum M.G., Kurths J.
Phys.rev. Lett. 1997. V. 55. P.2353.
In the work, on the basis of the built-in mathematical model [17] Pikovsky A.S., Rosenblum M.G., Kurths J.
of the Bernoulli-Euler nanobeam, the influence of taking into Synhronization: a Universal Cocept in Nonlinear Sciences.
account the size-dependent coefficients on the character of the Cambridge University Press, 2001.
contact interaction is studied. The necessity of taking into
account the theory of higher order for the study of NEMS
elements in the form of beams is shown. The visualization of
contact interaction characters based on the wavelet transform,
both for studying phase chaotic synchronization and for
studying changes in the oscillation spectrum over time, allows
us to more fully consider various aspects of contact interaction.
5. Appreciation
This work was supported by the Russian Science
Foundation, grant № 16-11-10138-П.
6. References
[1] Fu, Y., Zhang, J. Electromechanical dynamic buckling
phenomenon in symmetric electric fields actuated microbeams
considering material damping. Acta Mech. 212, (2010), 29–42.
[2] M. Moghimi Zand, and M. T. Ahmadian, “Static pull-in
analysis of electrostatically actuated microbeams using
homotopy perturbation method”, Appl. Math. Model. 34
(2010), 1032–1041.
[3] Jia, X.L., Yang, J., Kitipornchai, S.: Pull-in instability of
geometrically nonlinear micro-switches under electrostatic and
Casimir forces. Acta Mech. 218, (2011), 161–174.
[4] Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.,
1994. Strain gradient plasticity: theory and experiments. Acta
Metall. Mater. 42, 475–487.
[5] Nix, W.D. Mechanical properties of thin films. Metall.
Trans. A 20, (1989), 2217–2245.
[6] E. Mazza, S. Abel,J. Dual, Experimental determination of
mechanical properties of Ni and Ni-Fe microbars, Microsystem
Technologies 2 (4) (1996), 197-202
[7] Lam DCC, Yang F, Chong ACM, Wang J, Tong P
Experiments and theory in strain gradient elasticity. J Mech
Phys Solids 51: (2003) 1477–1508
[8] Q. Ma,D. R. Clarke, Size dependent hardness of silver
single crystals, Journal of Materials Research 10 (4) (1995)
853-863.
[9] Dell’Isola F, Della Corte A, Esposito R, Russo L. Some
cases of unrecognized transmission of scientific knowledge:
From antiquity to Gabrio Piola’s peridynamics and generalized
continuum theories. Generalized Continua as Models for
Classical and Advanced Materials: Springer; (2016). p. 77-128.
[10] Cosserat E, Cosserat F. Théorie des corps déformables:
Paris: Hermann et Fils, 1909.
[11] Mindlin RD. Second gradient of strain and surface-tension
in linear elasticity. Int J Solids Struct. (1965);1:417-38.
[12] Yang F, Chong ACM, Lam DCC, Tong P. Couple stress
based strain gradient theory for elasticity. Int J Solids Struct.
(2002) ;39:2731-43.
[13] Kantor B. Ya/, Bogatyrenko T. L. A method for solving
contact problems in the nonlinear theory of shells. Dokl.
Ukrain Academy of Sciences. Ser A 1; (1986). pp 18-21.
[14] Krysko V.A., Koch M.I., Zhigalov M.V., Krysko A.V.
“Chaotic Phase Synchronization of Vibrations of Multilayer
Beam Structures” Journal of Applied Mechanics and Technical
Physics. 2012. Т. 53. № 3. С. 451-459.
[15] J. Awrejcewicz, M.V. Zhigalov, V.A. Krysko-jr., U.
Nackenhorst, I.V. Papkova, A.V. Krysko, 'Nonlinear dynamics
and chaotic synchronization of contact interactions of multi-
layer beams', in: 'Dynamical Systems - Theory', Eds. J.
Awrejcewicz, M. Kaźmierczak, P. Olejnik, J. Mrozowski, TU
of Lodz Press, 2013, 283-292 (ISBN 978-83-7283-588-8)