=Paper=
{{Paper
|id=Vol-2485/paper14
|storemode=property
|title=Visualization of Transition's Scenarios from Harmonic to Chaotic Flexible Nonlinear-elastic Nano Beam's Oscillations
|pdfUrl=https://ceur-ws.org/Vol-2485/paper14.pdf
|volume=Vol-2485
|authors=Vadim Krysko,Irina Papkova,Ekaterina Krylova,Anton Krysko
}}
==Visualization of Transition's Scenarios from Harmonic to Chaotic Flexible Nonlinear-elastic Nano Beam's Oscillations ==
https://ceur-ws.org/Vol-2485/paper14.pdf
Visualization of Transition’s Scenarios from Harmonic to Chaotic
Flexible Nonlinear-elastic Nano Beam's Oscillations
V.A. Krysko1, I.V. Papkova1, E.Yu. Krylova2, A.V. Krysko1
tak@sun.ru| ikravzova@mail.ru | Kat.krylova@bk.ru| anton.krysko@gmail.com
1
Saratov State Technical University, Saratov, Russia;
2
Saratov State University, Saratov, Russia;
In this study, a mathematical model of the nonlinear vibrations of a nano-beam under the action of a sign-variable load and an
additive white noise was constructed and visualized. The beam is heterogeneous, isotropic, elastic. The physical nonlinearity of the
nano-beam was taken into account. The dependence of stress intensity on deformations intensity for aluminum was taken into account.
Geometric non-linearity according to Theodore von Karman’s theory was applied. The equations of motion, the boundary and initial
conditions of the Hamilton-Ostrogradski principle with regard to the modified couple stress theory were obtained. The system of
nonlinear partial differential equations to the Cauchy problem by the method of finite differences was reduced. The Cauchy problem
by the finite-difference method in the time coordinate was solved. The Birger variable method was used. Data visualization is carried
out from the standpoint of the qualitative theory of differential equations and nonlinear dynamics were carried out. Using a wide
range of tools visualization allowed to established that the transition from ordered vibrations to chaos is carried out according to the
scenario of Ruelle-Takens-Newhouse. With an increase of the size-dependent parameter, the zone of steady and regular vibrations
increases. The transition from regular to chaotic vibrations is accompanied by a tough dynamic loss of stability. The proposed method
is universal and can be extended to solve a wide class of various problems of mechanics of shells.
Keywords: visualization of scenarios of transition of vibrations into chaos, geometric nonlinearity, nano-beam, inhomogeneous
material, micropolar theory, Euler-Bernulli model.
investigated [4]. It was found that the scale parameter strongly
1. Introduction affects the propagation of waves in nano-rods.
Various modified models of nano-beams were proposed for
Visualization behavior of elements of
the study of bending using non-local mechanics of solid [3, 17,
microelectromechanical systems (MEMS) in the form of beams,
16, 20]. More recently, a beam model based on a nonlocal
plates and shells under the action of various kinds of gradient stress theory was proposed in [10, 11] to study the
loads currently is an actual scientific problem [1,2,7,8,9,18].
mechanical behavior of inhomogeneous nano-beams.
One of the ways of MEMS evolution is to reduce their
The article [15] is devoted to the development of a linear
mechanical components to the nanoscale and to reduce their
theory for the analysis of the behavior of beams based on the
mass. Already now about 10% of GDP in European countries is mechanics of a micropolar continuum. The nature of bending
directly related to micro and nanoengineering. It is expected
and longitudinal waves in a micropolar beam of infinite length
that the potential development of traditional microelectronics
was investigated. The deformation of a cantilever beam under
will be exhausted in the coming decade. Further development of
the action of a transverse concentrated load on the free edge
electronics is associated with the development of
was also studied. In [13], the size-dependent behavior of
nanotechnology. The scope of the NEMS is very wide.
Timoshenko beams using a combination of micropolar theory
Nanosensors (cantilevers, nano-suspensions, resonators, etc.)
with nonlocal elasticity was modeled. The authors of [14]
and nanoactuators (nano-motors) are used in physics, biology,
proposed a new numerical approach to the analysis of the
chemistry, medicine (diagnostics, cellular nano- and
bending of Euler – Bernoulli nano-beams in the context of
microsurgery, drug delivery, the affected area of the body),
integral non-local models. The authors of [12] studied the
electronic industry, criminology. Despite all the advances in
nonlinear vibrations of functionally graded nano-beams based
nanotechnology, any work at the molecular level remains
on an elastic base and subjected to a uniform increase in
extremely complex scientific problem. NEMS operate on the
temperature. The effect of small size, which plays a significant
basis of other physical laws than MEMS.
role in the dynamic behavior of nano-beams, is considered here
The classical theory of continuum mechanics is not ideal for
using an innovative non-local integral model. The main partial
analyzing the dynamic properties of nanostructures since size
differential equations of the theory of Bernoulli-Euler beams
effects significantly affect the dynamic behavior of
using von Karman relations were obtained.
nanostructures. For objects with ultra-small dimensions, it is
The review of articles confirms the need to visualize the
necessary to use more complex theories, for example, modified
behavior of NEMS components using methods of nonlinear
couple stress theory, nonlocal theory of elasticity, surface
dynamics and taking into account the specifics of small sizes of
theory of elasticity, etc. A very important scientific problem is
NEMS objects and the operating conditions of NEMS.
taking into account the heterogeneity of the material when
Basically, the visualization of the frequency spectrum of the
designing the NEMS element, i.e. Dependencies of physical
signal is carried out only using the Fourier spectrum. Use for
properties of a material on deformation, spatial coordinates and
visualization of modal, phase portraits, cross-section, Poincare
time There are a large number of studies devoted to the size-
mapping, spectrum of Lyapunov exponents, etc. will allow you
dependent behavior of beams. Longitudinal vibrations of to explore the phenomenon of determinate chaos, to determine
heterogeneous rods at nano-dimensional levels using , nonlocal
its truth. For a qualitative assessment of the transition of the
theory of elasticity were studied [6]. It was shown that the
oscillations of the nano-system into chaos, it is necessary to use
heterogeneity of the material can strongly influence the
tools like Fourier analysis and wavelet analysis, based on the
longitudinal vibrations of nano-rods. Depending on the value of strengths of each of them, when visualizing. The methods
the coefficient of elastic modulus, the natural frequency of
proposed in this work for visualizing the behavior of NEMS
nano-rods may decrease or increase with an increase in the
elements can be useful in non-destructive testing systems, as
number of degrees of freedom [19]. In addition to the
well as for designing NEMS.
longitudinal vibrations of nano-rods, wave propagation was also
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
1. Problem statement e
i s 1 exp i
In the modified couple stress theory [21], the stored strain es
energy П for an elastic body with infinitely small strains is es and s – strain intensity and stress yields depending from
written as
1 longitudinal (x) and transverse (z) coordinates.
П ij ij mij ij d dA, (1) Equations of motion, the boundary and initial conditions of
2 A the beam follow from the Hamilton – Ostrogradski variational
where: ij components of the strain tensor and ij principle.
1
components of the symmetric curvature gradient tensor, which k1 2 dA, k3 ( 2 ) z 2 l 2 dA, b1 dA
are defined as follows: A A 2 A
ij mm ij 2 ij , (2)
h
1 2 2 u 1 u
2
ij u, 2
2 zdz 10 30
2 x x 2 x
h
(3) 2
1
ij , ij rot (u ) i , mij 2 l ij ,
1 2
h2
2 1 2 2u30
( 2 ) z 2 l dz x 2 q
2 2
here u - displacement vector with components ui , h
2
i x, y, z , θ is an infinitely small rotation vector with h
h
u10 2 u10 1 u30 2 2 2u30 (4)
2 dz
2 zdz
2
components i . Denote ij , ij , mij and ij respectively x x h
x 2 x h x
2 2
components: classical stress tensor σ , strain tensor ε , deviator g 2u30 g u
h h 30 ;
part of a symmetric tensor moment higher order m and the t 2 t
symmetric part of the curvature tensor χ; h h
2 u10 1 u30 2 2 2u30 g 2u10
E x h x 2 x h
E 2 dz 2 zdz 2
h .
, - Lame parameters, which also x t 2
(1 )(1 2 ) 2(1 ) 2 2
The system (4) is reduced to dimensionless form
depend on the coordinates and intensity of deformations; ij -
x z
Kronecker symbol. The parameter l appearing at the moment using the following dimensionless parameters: x , z ,
of higher order mij is an additional independent material a h0
parameter of length. It is connected with the symmetric tensor u30 a4 h b au E
of a rotation gradient. In this model, in addition to the usual u 30 , q q, h , b , u10 210 , E ,
G h0
4
h0 h0 b0 h0 G
Lame parameters, one more scale parameter of length l [7]
must be taken into account. This is a direct consequence of the
, where q q0 Sin pt -
h0 Gb0 g h02 Gb0 g
fact that in couple stress theory, the strain energy density t t,
function is a function of the strain tensor and the symmetric a2 a2
curvature tensor. It does not explicitly depend on rotation (the
load acting on the beam, q0 and p - amplitude and load
asymmetric part of the strain gradient) and the asymmetric part
of the curvature tensor. frequency, respectively.
Consider a beam of length a , constant thickness h . The
To system (4) are added boundary conditions rigid fixation
h h
beam occupies the area ( x, z ) | 0 x a, z . We
2 2 u30 x, t
u30 ( x, t ) u10 x, t 0, x 0; a (5)
introduce the notation: h0 – beam thickness in the center, b0 - x
beam width in the center, u30 x, t – bend deflection, u10 x, t and initial conditions
– midline displacement. Beam designed from isotropic but u30 x, t u10 x, t
heterogeneous material E( x, z, ei ) and ( x, z, ei ) – module u30 ( x, t ) u10 x, t 0, t 0 (6)
t t
elasticity and Poisson's ratio, depending on the coordinates and
intensity of deformations ei , according to the deformation
2. Solution methods
theory of plasticity, – coefficient damping, – unit
weight of the material, g – acceleration gravity. Euler- The integration of equations (4) with the boundary (5) and
Bernoulli hypothesis was applied. initial (6) conditions is carried out by the finite difference
Geometric nonlinearity is taken into account according to method. To improve accuracy, central difference schemes for
the Karman model. derivatives have been applied.
Nano dimension is taken into account by the modified The convergence of the method along the spatial coordinate
couple stress theory. was studied. To obtain results with the required degree of
To account for the physical nonlinearity of the material of accuracy, it suffices to split the integration interval [0, 1] into
beams, the deformation theory of plasticity and the method of 120 parts. It is necessary to solve an extensive system of
variable parameters of elasticity are used [1]. Diagram of equations. At each time layer, the iterative method procedure
variable parameters Birger's elasticity [5] was built. The value
deformations material i ( i ) can be arbitrary, but in numerical
of the Young's modulus in the spatial grid x, z was refined.
examples it is accepted for pure aluminum in the form:
The stability of the solution in time, i.e. the choice of the time
step is carried out according to the Runge principle.
The reliability of the numerical results is proved by the The transition from harmonic vibrations to chaotic is
complete coincidence of the solutions obtained by the method obtained according to the scenario of Ruel-Takens-Newhouse
described above with the results obtained using the Bubnov with l 0; 0.1; 0.3; 0.5 . Thus, an independent frequency and
form finite element method in the spatial coordinate. Then the linear combinations of the excitation frequency and independent
Cauchy problems were solved using the Runge – Kutta type frequency appear.
method from the second to the eighth order of accuracy. Visualization of the nature of vibrations at the amplitude
of the load q0 84 (Table 1), based on the power spectrum, it
3. Numerical results
shows harmonic vibrations. However, visualization based on a
Visualization of the calculation results for a rigidly clamped phase portrait demonstrates the presence of additional
geometrically and physically nonlinear beam, to which a frequencies, since the phase portrait has a thickening. For a
uniformly distributed alternating load with a frequency qualitative analysis of the behavior of the system, several
p 5.1 is applied, which coincides with the frequency of visualization tools must be considered together. With increasing
linear natural vibrations, was based on the methods of nonlinear load amplitude, the Fourier spectrum reflects the appearance of
dynamics. The parameters of the experiment: the material is harmonics in the signal at independent frequencies. With a
aluminum, the ratio of length to thickness 𝜆=50. further increase in the amplitude, the power spectrum shows a
The aim of the study was to visualize the scenarios for the continuous pedestal, and the phase portrait shows a solid spot.
transition of vibrations of nano-beams from harmonic to Which indicates the chaotic state of the system.
chaotic, depending on load changes. As well as visualizing the With increasing l the load value at which the transition
impact on accounting scenarios moment stresses, i.e. the value to chaotic vibrations takes place, because flexural rigidity of the
of an additional parameter l related to the tensor gradient system increases.
curvature χ .
4. Conclusion
To achieve this goal, the following visualization tools were
used: signals, phase and modal portraits, Poincaré sections and This study presents a visualization of nonlinear vibrations
maps, autocorrelation function, Fourier spectra, 2D and 3D of a flexible, inhomogeneous, rigidly clamped at the ends of a
wavelet spectra constructed on the Morlet mother wavelet were nano-beam under the action of a uniformly distributed
analyzed. Also changes the sign of the the largest Lyapunov alternating load. Using the Fourier spectrum and phase portrait
exponent (LLE) in time, depending on the value of the as a means of visualization, it was possible to determine the
additional independent length parameter l , was studied. transition from harmonic to chaotic vibrations according to the
At the table shows the most informative results for l 0 . scenario of Ruel-Tackens-Newhouse. Accounting moment
Table 1. Scenario of transition to chaos at l 0 . stresses and an increase in the value associated with this
parameter does not change the scenario of the transition of
q0 system vibrations to chaos.
Fourier spectrum Phase portrait
5. Acknowledgement
This work was supported by the Russian Science
Foundation project RNF № 19-19-00215.
84
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