=Paper=
{{Paper
|id=Vol-2744/short11
|storemode=property
|title=Chaotic Oscillations Type of Flexible Closed Cylindrical Nanoshells Under the Strip Loads Action (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2744/short11.pdf
|volume=Vol-2744
|authors=Tatiana Yakovleva,Michael Stasuk,Anton Krysko
}}
==Chaotic Oscillations Type of Flexible Closed Cylindrical Nanoshells Under the Strip Loads Action (short paper)==
https://ceur-ws.org/Vol-2744/short11.pdf
Chaotic Oscillations Type of Flexible Closed Cylindrical
Nanoshells Under the Strip Loads Action*
Tatiana Yakovleva[0000-0003-3238-2317] , Michael Stasuk, and
Anton Krysko[0000-0002-9389-5602]
Yuri Gagarin State Technical University of Saratov, Saratov, Russia
yan-tan1987@mail.ru, stasmikhail@yandex.ru,
anton.krysko@gmail.com
Abstract. The theory and data visualization of flexible closed cylindrical
nanoshells nonlinear dynamics under the strip loads action are constructed. The
theory is based on hypotheses: Kirchhoff-Love, modified couple stress theory,
geometric non-linearity adopted in the T. von Karman form. To obtain the de-
sired differential equations, the Hamilton-Ostrogradsky principle was used,
which makes it possible to obtain the desired differential equations in mixed
form describing nano effects. For reduction to the Cauchy problem in spatial
coordinates, the Bubnov-Galerkin method in higher approximations is applied.
Further, the Cauchy problem is solved by methods such as Runge-Kutta and
Newmark. The convergence of the Bubnov-Galerkin method is studied depend-
ing on the number of terms in the original functions expansion in spatial coor-
dinates. The oscillations transition scenario from harmonic to chaotic depending
on the number of series members in the Bubnov-Garekin method, as well as de-
pending on the type of load, geometric and size-dependent parameters, is inves-
tigated. The numerical experiment results were visualized by nonlinear dynam-
ics methods and using wavelet analysis. It was revealed that the oscillations
type substantially depends on these parameters; two types of chaos are ob-
served: chaos and hyperchaos. This was revealed according to the chaos criteri-
on given by Gulik, and the Lyapunov exponents study by the methods of
Rosenstein, Kantz, and Wolf. A chaos type analysis was carried out based on
the signs of Lyapunov exponents spectrum calculated by the Sano-Sawada
method.
Keywords: Data Visualization, Flexible Closed Cylindrical Nanoshells, Modi-
fied Couple Stress Theory, Chaos, Hyperchaos, Lyapunov Exponents.
1 Introduction
The nonlinear dynamics problems visualization allows a qualitative assessment of the
complex mechanical structures behavior at a new level. The modern technology con-
stituent elements are subject to external dynamic action of a forceful nature. This fact
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons Li-
cense Attribution 4.0 International (CC BY 4.0).
* This work was supported by the RFBR according to the research project № 20-08-00354.
2 T. Yakovleva, M. Stasuk, A. Krysko
necessitates a comprehensive study of the structures behavior [1-3], determining their
limiting states and identifying the type of chaotic oscillations. Previously, the chaotic
oscillations type analysis was mainly considered in physics and radiophysics [4]. In
mechanics, the type of chaotic state was studied using classical systems as an example
[5-6]. An important issue is the methods of the results scientific visualization, which
allow obtaining reliable information about the ongoing process. This article is aimed
at constructing a theory and data visualization for the chaotic oscillations analysis of
flexible closed cylindrical nanoshells under the strip loads action.
2 Mathematical Model of a Cylindrical Nanoshell
The study object is a closed
cylindrical shell, occupying a region
in the space R3, where L and h – are shell
Ω = {x, y, z | ( x, y, z ) [0; L] [0; 2 ] [−h; h]}
length and thickness. The coordinate system is entered as shown in Fig. 1.
Fig. 1. Scheme of closed cylindrical nanoshell.
The shell is isotropic, homogeneous, elastic and obeys Hooke's law. Geometric non-
linearity is taken into account according to the T. Karman model. The direct normals
hypothesis is valid (Kirchhoff-Love hypothesis). Given the modified couple stress
theory [7-8], we obtain the resolving equations of shell motion, boundary and initial
conditions from the Ostrogradsky-Hamilton variation principle. According to this
principle, close movements are compared that bring the system of material points
from the initial position at time t 0 to the final position at time t1 . For true move-
ments, the condition must be satisfied:
t1
( K − U + W ) dt =0 (1)
t0
Chaotic Oscillations Type of Flexible Closed Cylindrical Nanoshells Under the Strip Loads… 3
Here K - kinetic energy, U - potential energy, W - external work related to distributed
forces and energy dissipation.
Using the standard procedure of the variations calculus, we obtain the shell motion
equations and the deformations compatibility equation in the form:
2 2
D2 w− k y F − L ( w, F ) + w + w = q ( x, y ,t ),
x 2 t 2 t
(2)
1 2 1 2
w
F =− L ( w, w) − k y ,
Eh 2 x2
Eh3 Eh 2 2 4 () 4 () 4 ()
where D = + , () = +2 + ,
12(1− 2 ) 2(1+ ) x 4 2
x y 2 y 4
2 2 2 2
2w 2 F 2w 2 F 2w 2 F w w w
L( w, F ) = + −2 , L ( w, w ) = 2 − - are known non-
x2 y 2 y 2 x2 xy xy 2 2 xy
x y
linear operators.
We attach to the system (2) the boundary conditions for articulated support at the
ends:
2w 2F
w = 0; = 0; F = 0; = 0 at x = 0;1 , (3)
x 2 x 2
and periodicity condition at y=0;2π.
In addition to the boundary conditions, we add to the equations (2) the initial condi-
tions:
w |t =0 = 0, w |t =0 = 0 (4)
Equations (2-4) are reduced to dimensionless form using the following parameters:
2 h Eh 4 LR
w = hw , F = Eh F , t = t0 t , = / , x = Lx , y = Ry , k y = k y , q=q 2 2 = h ,
R 2 L R Eg
L
= , where L and R = R y – are shell length and radius. Here t - time, - re-
R
sistance coefficient of the medium in which the shell moves, = 0.3 – Poisson's ratio,
E - Young's modulus, g - acceleration of gravit, - material density, - size-
dependent coefficient, q( x, y, t ) – strip load. The line over dimensionless quantities is
omitted for simplicity.
In this paper, the chaotic oscillations type of a closed cylindrical size-dependent
nano-shell with pivotally fixed edges is first studied. A transverse distributed strip
alternating load with amplitude q 0 acts on the shell (see Fig. 2).
4 T. Yakovleva, M. Stasuk, A. Krysko
Fig. 2. Shell loading diagram.
3 Solution Methods
Systems of nonlinear partial differential equations are reduced to a system of ordinary
differential equations by the Bubnov-Galerkin method in higher approximations. The
Cauchy problem is solved by the Runge-Kutta method of the fourth accuracy order
and Newmark method.
Further visualization of the closed cylindrical size-dependent shell oscillations na-
ture study was carried out by nonlinear dynamics methods with the w signals con-
struction at the center of the applied shell load, phase portraits, Fourier power spectra
and using wavelet analysis. For the data visualization reliability, the Morlet, Gauss 8 -
Gauss 32, and Haar wavelets were used as the mother wavelet. Methods and ap-
proaches of nonlinear dynamics find their application in various branches of modern
science, in mechanics, history [9], in the analysis of EEG data (time series) [10], etc.
An analysis of the nanoshell chaotic oscillations type is carried out according to the
chaos criterion given by Gulik [11], as well as on the basis of the Lyapunov expo-
nents spectrum signs calculating by the Sano-Sawada method [12].To confirm the
reliability of the results obtained, the senior Lyapunov indices are calculated by sev-
eral methods: Wolf [13], Kantz [14], Rosenstein [15] and Sano-Sawada. The results
visualization was carried out using the software packages MATLAB and Mathcad.
4 Numeric Results
The shell is affected by a transverse distributed strip alternating load with amplitude
q0 , natural vibration frequency p = 20.3 , load angle 0 = 5.98 along the entire length
of the cylinder. In the Bubnov-Galerkin method, the series members number was
taken equal to N = 5, 7, 9, 11, 13, 15 for each of the size-dependent coefficient
values = 0, 0.1, 0.3, 0.5, 0.7 .
Chaotic Oscillations Type of Flexible Closed Cylindrical Nanoshells Under the Strip Loads… 5
Table 1. The dynamic characteristics of a cylindrical shell at = 1 , q0 = 0.18575 , N = 15 .
6 T. Yakovleva, M. Stasuk, A. Krysko
Table 1 presents the oscillations graphs of the nano shell for the following parameters
values = 1 , q0 = 0.18575 , N = 15 , = 0, 0.1 .
In the case of a macro-sized ( = 0 ) cylindrical shell, in addition to the driving os-
cillations frequency ωp, the Hopf bifurcation at p / 2 is observed in the Fourier
power spectrum. In the Lyapunov exponents spectrum, we have positive senior and
positive second Lyapunov exponents, while the third or sixth exponents have negative
values. This means that the oscillatory system is in a state of hyper chaos [5] (Table
1).
In the case of a nano scale ( = 0.1 ) cylindrical shell, in addition to the driving os-
cillations frequency ωp, the Hopf bifurcation ωp /2 and a number of frequencies are
observed in the Fourier power spectrum. In the Lyapunov exponents spectrum we
have a positive senior exponent, while the other five exponents are negative. This
indicates the chaotic nature of shell vibrations [5] (Table 1).
Table 2. The oscillation types of the cylindrical shell.
In the case of the size-dependent parameter 𝛾, at the previous remaining parameters
values the cylindrical nanoshell performs harmonic oscillations, since the senior Lya-
punov exponent has a value close to zero, and the remaining five exponents in the
spectrum have negative values [5] (Table 2).
Table 2 presents the visualization results of the chaotic oscillations type analysis
for a closed cylindrical size-dependent shell depending on the geometric parameter (
= 1 and = 3 ), size-dependent parameter ( = 0, 0.1, 0.3, 0.5, 0.7 ), series members
number in the Bubnov-Galerkin method (N = 5, 7, 9, 11, 13, 15). The each sign of the
Chaotic Oscillations Type of Flexible Closed Cylindrical Nanoshells Under the Strip Loads… 7
first six Lyapunov exponents in the spectrum is indicated. For = 1 , with an increase
in the size-dependent parameter 𝛾, chaotic oscillations transform into harmonic ones.
At = 3 such a process is observed only at N = 5, 15 .
5 Conclusion
In the work, by visualizing scientific data, a chaotic oscillations type analysis of flexi-
ble closed cylindrical nanoshells under the strip loads action is carried out. The scien-
tific visualization of the results is based on the construction of signals, phase portraits,
Fourier power spectra and the use of wavelet transforms. It has been established that
the Morlet wavelet is the most informative for this class of problems, since it gives
the best frequency localization at every moment in time. It is worth noting that the
Fourier power spectrum gives a general picture of the nature of the oscillations of
nanoshells over the entire time interval. The proposed approach allows us to study the
nonlinear dynamics of the flexible closed cylindrical nanoshells, under the influence
of an external alternating load, depending on the size-dependent coefficient. It was
revealed that the oscillations type significantly depends on the amplitude and frequen-
cy of the load, size-dependent and geometric parameters, as well as on the members
number of series in the Bubnov-Galerkin method. The cylindrical shell can perform
harmonic or chaotic oscillations, while depending on the number of positive Lyapun-
ov exponents, two types of chaos are observed in the spectrum: chaos and hyperchaos.
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