=Paper=
{{Paper
|id=Vol-2485/paper22
|storemode=property
|title=Investigation of the Effect of Additive White Noise on the Dynamics of Contact Interaction of the Beam Structure
|pdfUrl=https://ceur-ws.org/Vol-2485/paper22.pdf
|volume=Vol-2485
|authors=Olga Saltykova,Alexander Krechin
}}
==Investigation of the Effect of Additive White Noise on the Dynamics of Contact Interaction of the Beam Structure==
https://ceur-ws.org/Vol-2485/paper22.pdf
Investigation of the Effect of Additive White Noise on the Dynamics
of Contact Interaction of the Beam Structure
Оlga Saltykova1, Alexander Krechin1
olga_a_saltykova@mail.ru| san9.antonov@yandex.ru
1
Yuri Gagarin State Technical UniversityofSaratov, Saratov, Russia
The purpose of this work is to study and scientific visualization the effect of additive white noise on the nonlinear dynamics of beam
structure contact interaction, where beams obey the kinematic hypotheses of the first and second approximation. When constructing a
mathematical model, geometric nonlinearity according to the T. von Karman model and constructive nonlinearity are taken into account.
The beam structure is under the influence of an external alternating load, as well as in the field of additive white noise. The chaotic
dynamics and synchronization of the contact interaction of two beams is investigated. The resulting system of partial differential
equations is reduced to a Cauchy problem by the finite difference method and then solved by the fourth order Runge-Kutta method.
Keywords: nonlinear dynamics, contact interaction, chaotic phase synchronization, white noise.
i 1, 2 - are serial number of beams.
1. Introduction
u i wi u i wi
2 2
The mechanics of contact interaction is one of the most F1 (u i , wi ) ,
rapidly developing topics of the mechanics of a deformable solid x 2 x x x 2
2 wi wi
2
and is widely used in various fields of science [2, 4, 5]. A 3 2 wi wi
mathematical model of the contact interaction of two beams, F2 ( wi , wi ) , F3 ( wi , wi ) ,
described by the kinematic hypotheses of the first and second
2 x x
2
x 2 x
2
approximations [1], was constructed. An external alternating 2 wi ui 2 wi wi
load and a white noise field affect one of the beams. Using the L1 ( wi , ui ) , L2 ( wi , wi ) ,
x x
2
x 2 x
means of scientific visualization of the results of mathematical
modeling, the nonlinear dynamics of the contact interaction of wi 2 ui w 2 wi
L3 (wi , ui ) , L4 ( wi , wi ) i are the
the beam structure located in the field of additive white noise is x x 2
x x 2
studied.
nonlinear operators, xi -is lateral shift function, wi , ui – are
2. Statement of the problem functions of deflection and displacement of beams, respectively,
К– stiffness coefficient of transversal compression of the
Geometric nonlinearity of beams was adopted according to structure in the contact zone, hk – the gap between the beams,
the model of T. von Karman, the contact interaction is described
by the B.Ya.Kantor model [3]. The equations of motion, the thickness of the beams b 1 , 1 - damping coefficient,
boundary and initial conditions are obtained from the Hamilton- a
Ostrogradsky energy principle. Beam 1 obeys the kinematic - beam geometry parameter.
hypothesis of the first approximation (Euler-Bernoulli model) 2h
under the action of transversal load and white noise, beam 2 is The boundary conditions in the case of rigid pinching and the
described by the kinematic hypothesis of the second initial conditions should be added to equations (1).
approximation (Timoshenko model). The study of nonlinear For the beam described by the hypothesis of the first
dynamics is based on the study of phase portraits, wavelet and approximation, the boundary conditions (2) and the initial
Fourier spectra, signals, chaotic phase synchronization, conditions (3):
Lyapunov indicators. The values of the highest Lyapunov wi (0, t ) wi (1, t ) ui (0, t )
exponent are calculated by three methods: using the Kantz, Wolf w 0, t wi 1, t (2)
and Rosenstein algorithm. ui (1, t ) i 0.
x x
The equations of beams motion will take the form:
wi ( x) t 0 0, ui ( x) t 0 0,
1 1 4 w1 2 w1
2 F2 ( wi , wi ) F1 ui , wi
w
1 1 wi x ui x (3)
12 x 4 t 2 t 0, 0.
t t 0 t t 0
(1) K ( w1 w2 hk ) q( x, t ) 0,
i
2 For the beam described by the hypothesis of the second
u1 F w , w u1 0;
2
approximation, the boundary conditions (4) and the initial
x 2 3 i i
t 2 (1) conditions (5):
1 w2 x 2 w(0, t ) w(1, t ) 0; u (0, t ) u (1, t ) 0;
1 L1 ( wi , ui ) 3 L2 ( wi , wi ) L3 ( wi , ui )
2
2 w0, t w1, t
3 x x 2 2 (4)
x (0, t ) x (1, t ) 0; 0;
x x
(1)i K ( w w h ) 2
w w
2
1 2 0; w( x, t ) t 0 0, u ( x, t ) t 0 0, x ( x, t ) |t 0 0,
1 2 k
t 2 t
2
u2 u2
2
wx, t u x, t x, t (5)
x 2 L4 ( wi , wi ) t 2 0; 0, 0, x 0.
t t 0 t t 0 t |t 0
2 x 2 w x2
2
Beam 1 is affected by a distributed transverse alternating load
82 2 x 2 0; i 1, 2,
x x t 2
2 of the form, additive white noise is added to the system of
equations in the form of a random term with constant intensity
Pn Pn0 (2.0 * rand() /(65535 1.0) , Pn0 — is the noise
intensity; rand() — standard C++ function that accepts a random
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
integer value from 0 to 65535. This model was calculated using a Increasing the load to 1400 (Table 1) leads to a change in the
program written in C++. Visualization and analysis of the results frequencies of the beams, on the power spectrum of the beam 1
was carried out on the basis of the MathCad and MATLAB there is one frequency: p 5,1 . On the power spectrum of
programs.
beam 2, there are no pronounced frequencies.
q q0 sin( t ) Pn , (6) The oscillations of the system at a given load are harmonic,
where - is load frequency; q - is load amplitude; Pn - chaos is not observed, as evidenced by phase portraits, wavelet
random term with constant intensity. The resulting system of spectra portrait of phase synchronization, as well as Lyapunov
partial differential equations is reduced to an Ordinary exponents, calculated by three different methods (Wolf,
Differential Equation system by the finite difference method Rosenstein, Kantz) are negative.
with a second-order approximation. The obtained Cauchy Table 2
problem is solved by the Runge-Kutta method. Dynamic characteristics of beams
50 , hk 0,1, q0 1400, Pn0 1
3. Results of a numerical experiment Beam 1 Beam 2
Power Phase portrait Power Phase
We present the results of a study of the nonlinear dynamics
spectrum spectrum portrait
of contact interaction of a beam structure in a white noise field,
where beam 1 is described by the Euler – Bernoulli hypothesis,
beam 2 is subject to Timoshenko’s hypothesis (Fig. 1).
Wavelet spectrum Wavelet spectrum
Figure 1. Beams structure
Table 1
Dynamic characteristics of beams Phase synchronization
50 , hk 0,1, q0 1400, Pn0 0
Beam 1 Beam 2
Powerspectru Phase portrait Powerspectru Phase
m m portrait
Wavelet spectrum Wavelet spectrum
Lyapunov exponents
1.Wolf=0,00421 1.Wolf=0,00456
2.Rosenstein=0,05380 2.Rosenstein=0,05964
3.Kantz=0,01757 3.Kantz=0,03093
When adding a noise component (Table 2), the dynamics of
Phasesynchronization the structure changes.
The power spectrum of beam 1 contains five frequencies.:
p 4 p 5 p
1 , 2 0.757 , 3 ,4 , p 5,1 .Two
11 6 11
frequencies 2 , p - are linear independent, and other
frequencies are their linear combinations.
The power spectrum of beam 2 contains four frequencies.:
12 p 14 p
5 , 6 1.82, 7 , p 5,1 .Two frequencies
100 16
6 , p -are linear independent, and other frequencies are their
linear combinations.
When adding a noise component, the system went into a
chaotic state, which is visible in the wavelet spectra and in the
Lyapunov exponents phase synchronization portrait, as well as in Lyapunov’s
1.Wolf=-0,020421 1.Wolf=-0,020456 indicators.
2.Rosenstein=-0,075380 2.Rosenstein=-0,054964 The transition of the system to chaos occurred through the
3.Kantz=-0,021757 3.Kantz=-0,063093 scenario of Ruel-Takens-Newhouse.
The first contact of the beams occurs under load q0 =800.
Table 3 Table 4
Dynamic characteristics of beams Dynamic characteristics of beams
50 , hk 0,1, q0 55000 , Pn0 0 50 , hk 0,1, q0 55000, Pn0 1
Beam 1 Beam 2 Beam 1 Beam 2
Power Phase portrait Power Phase Powerspectru Phase portrait Powerspectru Phase
spectrum spectrum portrait m m portrait
Wavelet spectrum Wavelet spectrum Wavelet spectrum Wavelet spectrum
Phase synchronization Phasesynchronization
Lyapunov exponents Lyapunov exponents
1.Wolf=0,00926 1.Wolf=0,00875 1.Wolf=0,00926 1.Wolf=0,00875
2.Rosenstein=0,04083 2.Rosenstein=0,05534 2.Rosenstein=0,04083 2.Rosenstein=0,05534
3.Kantz=0,04083 3.Kantz=0,02858 3.Kantz=0,04083 3.Kantz=0,02858
At q0 = 55000 (Table 3) the power spectrums of beam 1 and Table 5
beam 2 contains five frequencies: Dynamic characteristics of beams
p 3 p 50 , hk 0,1, q0 1400
10 , 11 2.12, 12 , 13 2( p 3 ), p 5,1 .
5 5
Signals
At q0 =55000, t>100 frequency synchronization occurs:
1 , 2 , p .
With an increase in the amplitude of the forced oscillations,
the character of the beam signals changes from quasi-periodic to
chaotic.
We can observe the scenario of Ruel-Takens-Newhouse.
Wavelet spectra visualization allow you to see the change in the
nature of oscillations of beams in time.
In Table 4, when adding white noise Pno=1, visual, and
therefore qualitative changes in the dynamics of the model were
not detected.
The power spectrums of beam 1 and beam 2 contains five
frequencies described above. Power spectrumW1 Power spectrumW2
Note that in this case the influence of the noise load
practically did not affect the nonlinear dynamics of the contact
interaction of the beams.
An increase in the amplitude of white noise does not lead to
a change in the scenario of transition of oscillations into chaotic.
In Tables 5 and 6, we compare the Fourier spectra and signals
without a white noise field and with noise, respectively.
Table6 6. References
Dynamic characteristics of beams
50 , hk 0,1, q0 55000 [1] Awrejcewicz, J., Krysko, A.V., Pavlov, S.P., Zhigalov,
M.V., &Krysko, V.A. (2017). Chaotic dynamics of size
Signals dependent Timoshenko beams with functionally graded
properties along their thickness. Mechanical Systems and Signal
Processing, 93, 415-430.
[2] Awrejcewicz, J., Krysko-Jr, V.A., Yakovleva, T.V.,
Krysko, V.A. (2016). Noisy contact interactions of multi-layer
mechanical structures coupled by boundary conditions. Journal
of Sound and Vibration, 369, 77-86.
[3] Kantor B.Ya. Contact problems of the nonlinear theory of
shells of revolution, Kiev, Naukova Dumka, 1991, p. 136
[4] Krysko, V.A., Awrejcewicz, J., Papkova, I.V., Saltykova,
O.A., Krysko, A.V. (2019). Chaotic Contact Dynamics of Two
Microbeams under Various Kinematic Hypotheses. International
Journal of Nonlinear Sciences and Numerical Simulation, 20(3-
4), 373-386.
Power spectrum W1 Power spectrum W2 [5] Yakovleva, T.V., Krysko Jr, V.A., &Krysko, V.A. (2019,
March). Nonlinear dynamics of the contact interaction of a three-
layer plate-beam nanostructure in a white noise field. In Journal
of Physics: Conference Series (Vol. 1210, No. 1, p. 012160). IOP
Publishing.
Visualization of signals and power spectra allows to visually
see (Table 5 and Table 6) the qualitative changes in the vibrations
of the beam structure, under the influence of an external
alternating load of different intensity and white noise.
4. Conclusion
A mathematical model of the contact interaction of two
geometrically non-linear beams, described by the kinematic
hypotheses of the first and second approximation, is constructed.
Data visualization made it possible to compare signals, phase
synchronization, phase portraits and identify features of the
dynamics of contact interaction of the studied beam structure.
One of the structure beams is under the influence of an external
distributed alternating load and in the field of white additive
noise. The effect of the intensity of the noise component (Pn) on
the amplitude-frequency characteristics of the beams was
investigated. A numerical experiment was performed for Pn =
0.1; 0.5; 1, with the same characteristics of the external
alternating load. With small amplitudes of forcing vibrations
(q0<10000), the presence of additive white noise with intensity
Pn = 1 significantly changes the nonlinear dynamics of the
structure under study and leads to a transition of system
oscillations from harmonic to chaotic. When Pn = 0.1;0.5 the
influence of white noise is not significant and can be neglected.
At q0> 12000, the effect of additive white noise is less obvious.
This is due to the fact that the system is already in a chaotic state.
The influence of additive white noise on the scenario of transition
from harmonic to chaotic oscillations is investigated. Using
scientific data visualization shown it is shown that the
consideration of the noise component does not affect the scenario
of transition of oscillations to chaotic ones. The transition to
chaotic oscillations occurs according to the scenario of Ruel-
Takens- Newhouse. The phenomenon of a decrease in the noise
component under the action of additive white noise was found
(Table 6).
5. Acknowledgments
This work was supported by the grant of the Russian Science
Foundation16-11-10138.