=Paper=
{{Paper
|id=Vol-2744/short7
|storemode=property
|title=Scientific Visualization of the Results of a Numerical Experiment of the Nonlinear Dynamics of a Nanoscale Beam Structure (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2744/short7.pdf
|volume=Vol-2744
|authors=Olga Saltykova
}}
==Scientific Visualization of the Results of a Numerical Experiment of the Nonlinear Dynamics of a Nanoscale Beam Structure (short paper)==
https://ceur-ws.org/Vol-2744/short7.pdf
Scientific Visualization of the Results of a Numerical
Experiment of the Nonlinear Dynamics of a Nanoscale
Beam Structure*
Olga Saltykova [0000-0002-3880-6662]
Yuri Gagarin State Technical University, 77 Politehnicheskaya st., Saratov, Russia
olga_a_saltykova@mail.ru
Abstract. The paper presents the results of scientific visualization of the
nonlinear dynamics of contact interaction of a nanoscale beam structure
under the action of an external harmonic load. The beam structure consists
of two beams obeying the kinematic hypotheses of Euler-Bernoulli and S.P.
Timoshenko. The constructed mathematical model takes into account
geometric and constructive nonlinearities. The size-dependent behavior of
the structure is implemented on the basis of the modified moment theory of
elasticity. The resulting system of partial differential equations is reduced
to a system of ordinary differential equations by the second order finite
difference method. The Cauchy problem is solved by the fourth order
Runge-Kutta method. In this work, using the methods of scientific
visualization of the results of applying the methods of nonlinear dynamics,
the influence of the size-dependent parameter and external load on the
vibrations of the beam structure is investigated. As methods for studying
nonlinear dynamics, the work uses wavelet spectra based on the mother
Morlet, Fourier power spectra, signals. The use of scientific visualization
methods makes it possible to develop specific recommendations for the
operating conditions of the beam structure. This, in turn, makes it possible
to avoid unwanted vibration modes of beam nanostructures, which are
widely used as sensitive elements of sensors of micro and nano
electromechanical systems.
Keywords: Nanobeams, Nonlinear Dynamics, Wavelet Analysis,
Timoshenko Model, Euler-Bernoulli Model, Modified Moment Theory of
Elasticity.
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).
* Supported by RFBR grant № 18-41-700001 r_a
2 O. Saltykova
1 Introduction
Scientific visualization is one of the effective methods of analysis of data obtained as a
result of numerical experiments, therefore, it is widely used in various kinds of research
[1], [2]. The main objective of this work is the application of scientific visualization
methods to study the nonlinear dynamics of a nanoscale beam structure.
Such beam structures are components of sensitive elements of various
nanoelectromechanical systems and sensors and during the operation of such systems
can be subjected to various external influences, including dynamic ones [3].
2 Formulation of the problem
The object of study is the beam structure, consisting of two nanobeams with a gap
between them (Fig. 1). Beam 1 is described by the Euler-Bernoulli kinematic
hypothesis, and beam 2 is described by the Timoshenko hypothesis, which takes into
account the shear strain. l - is a length, 2h - is a thickness of beams. Both beams are
geometrically non-linear according to the model of T. von Karman. A transverse
alternating load (1) distributed over the surface acts on the beam 1, and the beam 2 only
starts moving after contact with the beam 1.
q( x, t ) = q0 sin( p t ), (1)
where q( x, t ) - is the alternating load; p -is the frequency; q 0 - is the amplitude; t
- it is time.
Contact interaction is described by the model of Cantor B.Ya. [4], that is, at each
time step, the problem of contact of two bodies with an unknown contact boundary is
solved. The equations of motion, boundary and initial conditions are obtained from the
Hamilton-Ostrogradsky energy principle. Classical solid mechanics is not able to
interpret and predict size-dependent behavior, due to the lack of a parameter that takes
into account scale effects. Therefore, when obtaining the equations, we use the
modified moment theory of elasticity [5], where the scale parameter of the material
length is taken into account, taking into account the effect of higher-order
moments [6,7].
Fig. 1. Nanobeam structure
Scientific Visualization of the Results of a Numerical Experiment of the Nonlinear… 3
The equations of motion of beams in a dimensionless form take the form:
1 1 2 4 w1 2 w1 w
2 F2 ( wi , wi ) + F1 (u i , wi ) + − + 4
− − 1 1 +
+ t
x t
12 2 (1 ) 2
+ (−1) i K ( w1 − w2 − hk ) + q( x, t ) = 0,
2
u1 2 u1
2 + F 3 (w i , wi ) − = 0;
x t 2
1 2 w x 2 1
+ 3
2
+ L ( w , u ) + L2 ( wi , wi ) + L3 ( wi , u i ) + (2)
2 1 i i
x
3 2 x 2
2 w2 w
+ ( −1) i
K ( w 1 − w 2 − hk ) − − 1 2 = 0;
t 2 t
2u 2u2
2 2
+ L4 ( wi , wi ) − = 0;
x 2 t 2
2 2 x 2 64 w2 32 2
2 x2 3 w x 2
2
− + x2 + − − = 0; i = 1, 2,
x 2 1 + x 2(1 + ) x 2 x 3 t 2
2u1 w1 u1 2 w1
i = 1, 2 - is the serial number of beams. F1 (ui , wi ) = + ,
x 2 x x x 2
2
3 2 w1 w1 2 w1 w1 2 wi ui
F2 ( wi , wi ) = , F3 (wi , wi ) = , L1 ( wi , ui ) = ,
2 x 2 x x 2 x x 2 x
2
2 wi wi wi 2 ui w 2 wi
L2 ( wi , wi ) = , L3 (wi , ui ) = , L4 ( wi , wi ) = i are
x 2 x x x 2 x x 2
nonlinear operators, xi is the transverse shear function, wi , u i are the deflections and
displacements of the beams, respectively, hk is the gap between the beams (see Fig.
1), = 0.3 is the Poisson's ratio, is the scale parameter of the material length, 1 is the
l
dissipation coefficient, = is the geometrical parameter.
2h
To model contact interaction, the theory of B.Ya.Cantor [4], according to which it
is necessary to add a term to the equations of motion of structural elements
qk = (−1)i K1(w1 − hk − w2 ) , K is the stiffness coefficient of the transverse
compression of the structure in the contact zone. The function Ψ is defined by the
formula = 1 + sign(w1 − hk − w2 ) , (if w1 hk + w 2 , then there is contact
1
2
between the structural elements and Ψ = 1; in the absence of contact Ψ = 0).
4 O. Saltykova
To equations (2), boundary conditions in the case of rigid pinching (3), (5) and initial
conditions (4), (6) for beams 1 and 2, respectively, should be added.
For beam 1:
w1 (0, t ) w1 (1, t )
w1 (0, t ) = w1 (1, t ) = u1 (0, t ) = u1 (1, t ) = = = 0. (3)
x x
w1 (x ) u (x )
w1 ( x) t =0 = 0, u1 ( x) t =0 = 0, = 0, 1 = 0. (4)
t t =0 t t =0
For beam 2:
w2 (0, t ) = w2 (1, t ) = 0; u 2 (0, t ) = u 2 (1, t ) = 0;
w2 (0, t ) w2 (1, t ) (5)
x 2 (0, t ) = x 2 (1, t ) = 0; = = 0;
x x
w2 ( x, t ) t = 0 = 0, u 2 ( x, t ) t = 0 = 0, x 2 ( x, t )|t = 0 = 0,
w2 (x, t ) u (x, t ) (x, t ) (6)
= 0, 2 = 0, x 2 = 0.
t t =0 t t =0 t |t = 0
3 Solution methods
The resulting system of partial differential equations is reduced to an ordinary
differential equation system by the second-order finite difference method. The obtained
Cauchy problem is solved by the fourth-order Runge-Kutta method.
To study the nonlinear dynamics of the contact interaction of a nanoscale beam
structure on the basis of nonlinear dynamics methods, we construct signals, plots, power
spectra, wavelet spectra based on the Morlet mother wavelet, and phase portraits. The
totality of the visual presentation of the results obtained by these methods will allow us
to have an almost complete picture of the features of the nonlinear dynamics of the
described mechanical structure, depending on the value of the size-dependent parameter
and the amplitude of the external load q 0 .
4 Numerical experiment
We are present the results of a numerical experiment with the following values of the
control parameters: = 50 ; hk = 0.1; q 0 1000 ; 5000 ; = 1; 0.1; 0.3; 0.5.
Table 1 shows the signals of beams 1 (red) and 2 (blue) w(t , 0.5) , t (0;20 ) for
various values of the amplitude load q 0 1000 ; 5000 and the size-dependent
parameter 0.1; 0.3; 0.5.
Depending on the value of the size-dependent parameter , the oscillation
Scientific Visualization of the Results of a Numerical Experiment of the Nonlinear… 5
amplitude of the beam 1 changes, an increase in the parameter leads to a decrease in the
amplitude of the oscillations. Visualization of the beam signals allows you to see the
points in time at which the beam contacts. But, to determine the shape of the vibrations
of the beams, it is necessary to analyze the plots of the deflections of the beams.
Table 1. Signals of the nanobeams for q 0 1000 ; 5000 ; = 1; 0.1; 0.3; 0.5
Table 2 shows the plots for = 0.1 , q 0 1000 ; 5000 , at different points in time.
Graphical visualization of deflection diagrams of a beam nanostructure allows obtaining
information about the vibration mode at any moment of interest, which cannot be seen
with only a numerical solution.
6 O. Saltykova
Table 2. Plots of deflections of beams 1 and 2.
The diagrams at = 0.1 , q0 = 1000 , are given for the moment of the first contact
of the beams, at t = 3.264 . It was possible to determine the time of the first touch by
visualizing the beams deflection diagrams in three-dimensional space ( x, z , t ) . As
you can see, the first contact of the beams does not occur at the central point, but in
quarters. The increase in load leads to the fact that the first contact of the beams occurs
at t = 0.64 , and the appearance of the diagrams at the first contact changes. The
visualization of vibration modes allows you to develop specific recommendations for
the operating conditions of the studied mechanical structure.
To obtain the frequency characteristics of signals, we use the wavelet spectra based
on the Morlet mother wavelet, as well as the Fourier power spectra. Wavelet analysis
allows you to see the change in the frequency characteristics of the oscillatory process
in time and is a kind of "microscope".
Tables 3 and 4 show wavelets p (t ) and Fourier power spectra for beams 1 and 2
at q 0 = 1000 (table 3) and at q 0 = 5000 (table 4).
Scientific Visualization of the Results of a Numerical Experiment of the Nonlinear… 7
Table 3. Wavelet spectra at q0 = 1000 ; 0.1; 0.3; 0.5
Beam 1 Beam 2
= 0.1
Wavelet spectrum
Fourier Power Spectra
= 0.3
Wavelet spectrum
Fourier Power Spectra
= 0.5
Wavelet spectrum
Fourier Power Spectra
8 O. Saltykova
Table 4. Wavelet spectra at q 0 = 5000 ; 0.1; 0.3; 0.5
Beam 1 Beam 2
= 0 .1
Wavelet spectrum
Fourier Power Spectra
= 0.3
Wavelet spectrum
Fourier Power Spectra
= 0.5
Wavelet spectrum
Fourier Power Spectra
Scientific Visualization of the Results of a Numerical Experiment of the Nonlinear… 9
Wavelets are given for t (0;20 ) , and the Fourier power spectra are built on the time
interval t (0;1024 ) . This is due to the fact that if the wavelet spectra are plotted over
the entire time interval, the frequency changes characteristic of the initial time interval
are not visible.
The wavelet spectra of beams 1 and 2 are distinguished by the presence of a low-
frequency component in the spectrum of beam 1 = 0.5 when the wavelet spectra of
beam 2 is generated, a low-frequency component occurs over the entire time interval.
The Fourier power spectra do not differ in frequency components. The same frequencies
can be seen on wavelet spectrum.
Table 4 shows the wavelet spectra and the power spectra of the Fourier beams at
q0 = 5000 .
The wavelet spectra for reducible loads differ significantly for the same values of the
size-dependent parameter, as well as the Fourier power spectra. An increase in the size-
dependent parameter leads to a decrease in frequencies, i.e., to regularization of
oscillations. This is due to a decrease in the amplitude of oscillations of the beams,
which was mentioned above.
5 Conclusions
Scientific visualization of the nonlinear dynamics of the contact interaction of the beam
structure allows you to have a detailed idea of the influence of control parameters on
the nature of the vibrations of the beam structure, which makes it possible to develop
specific recommendations on the operating conditions of the beam structure and avoid
undesirable vibration modes, which can ultimately lead to the destruction of the beam
structure.
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