Synchronization in Chains of Material
Particles with Non-linear Features
Julijana D. Simonovic
Doctor of Tehnical Sciences
University of Nis
Faculty of Mechanical Engineering
The paper presents the marvelous possibilities of identical synchronization
in the dynamics of the chains of material particles. The nature of coupling
elements between particles has got damped, linear and non-linear elastic
properties. The one or more particles in the chain are periodically forced.
Depending of coupling elements features and the positions of excitations
the synchronization effect is less or more present. The multi-parameters
analyses were done by presentation of numerical simulation in the phase
space of output variables of coupled particles. Concluding remarks have
been considered about nature of coupling and symmetries in system of the
external excitation positions that are necessary and enough for reaching
the identical synchronization threshold.
Keywords: chain of material particles, coupling elements, identical
synchronization, synchronization threshold, , multi-parameters analyses.
1. INTRODUCTION
It would not be too much of an exaggeration to say that
oscillations are one of the main forms of motion. They
range from the periodic motion of planets to random
openings of ion channels in cell membranes. The chain
of material particles coupled with elastic and damped
elements periodically forced performs oscillatory
motions with all characteristic phenomena like as
dynamical absorption, resonance, beating, modulations,
synchronisations and could be used like a model of
various natural phenomena. Nature is not based on
isolated individual systems. It is rich in connections,
interactions and communications of different kinds that
are complex beyond belief. With this, synchronization is
the most fundamental phenomenon associated with
oscillations. Chains of material particles with non-linear
coupling are systems with pronounced nonlinear
characteristics. When one couple material particles that
oscillates a lot of things can happen. For example,
oscillations can stop altogether, which might be good
sometimes, but occasionally disastrous. Or they could
become totally unpredictable-but you might like it
nevertheless because it looks beautiful. But the
phenomenon which is most often associated with
synchronization is the change of the time scales of
interacting systems: if one couples the systems cleverly,
they can start to oscillate “syn-chronously,” which
means “sharing the same time” [1].
The nonlinearity brings the properties of
exponentially divergence of trajectories of particles that
starts from very close points in phase space,
nevertheless even in that systems on may find
synchronization, which is rather surprising detail [1-2].
In the general sense the synchronization consider the
correlation or mutual response in time behaviour of two
or more processes. In other words, regimes of
Received: May 2014, Accepted: November 2014
Correspondence to: Julijana Simonovic
University of Nis Mechanical Engineering Faculty ,
Medvedeva A. 14, 18000 Nis, Serbia
E-mail: [email protected]
doi:10.5937/fmet1404341S
© Faculty of Mechanical Engineering, Belgrade. All rights reserved
subsystems have to coincidence after some period of
time [3-4]. One can say that synchronization is the most
fundamental phenomenon that occurs in oscillating
processes. In most general terms, synchronization can
be defined as an adjustment of the time scales of
oscillations due to interaction between the oscillating
processes, [5].Investigation of dynamics of chains of
material particles in the systems with more than three
degrees of freedom, even in the field of classical and
linear chain forced dynamics, is important not only for
mechanical signal processing, but also for electrical
signal processing and signal filtering, for processing
biodynamical signals in life systems (DNA double helix
chains [6], biodynamical chain oscillators [7-8]) and
also for university teaching and integrations of scientific
results in different scientific fields.
This paper presents possibilities of identical
synchronization (IS) in the non-linear chains of material
particles. First of all we present model of
synchronization of two material particles coupled with
damped and linear and non-linear elastic elements that
is mathematically tractable. Our experience tells us that
models with the latter property are exceptional. While
the treatment of just a two coupled elements can be
done within the framework of non-linear dynamical
systems, the description becomes much more
complicated for a large number of such elements.
Considering that in the next section we use multiparametric numerical simulations by using the
possibilities of the Mathematica software for crossed
chains with nine material particles. Depending of values
of coupling coefficient and the positions of the external
excitation the effect of synchronization is less or more
present. After all, few conclusions about the nature of
coupling like as necessary strength of coupling
coefficients which is needed and enough for IS in the
particular chain systems were discussed.
From all different cases of synchronization the identical
synchronization (IS), as a particular case of general
synchronization, was been of interest in this paper as in
the paper [9]. The most simply case of IS could be bring
off when the particles are coupled with sufficient
FME Transactions (2014) 42, 341-345 341
coupling strength so that their states are equal after
transient changes.
The work was originally presented on the 1st
International Symposium on Machines, Mechanics and
Mechatronics (SMMM 2014),on July 1-2. 2014,
University of Belgrade, Faculty of Mechanical
Engineering,Serbia.
2. MATHEMATICAL MODELING
x1 t 
F1 t   F1 cos 1t  0 
m1
x 2 t 
b
c, c~
a1 t    1 a1 t  
F2 t   F2 cos  2t  0 
m2

1 t   p̂1  1  
a 2 t    2 a 2 t  
Mathematical modelling can assist in formulating
hypotheses and finding the experiments that may
differentiate between different mechanisms, can help in
determining which of the assumed mechanisms is likely
to be most important for the observed effects. Finally,
and perhaps most importantly, mathematical modelling
is the tool for synthesis-the process of combining
separate elements in order to form a coherent wholeallowing reconstruction of the function of phenomena as
a whole based on the acquired knowledge of elementary
processes.
The basic model of forced non-linear oscillations of
two material object with mass m1 and m2 that are coupled
with elements of damped and elastic properties, Fig.1.
The elastic coupling element has both linear and nonlinear parts, the non-linear part is introduced like a
nonlinearity of third order.
amplitudes and phases in the first asymptotic averaged
approximation [10] may be rewriten in the general form
as follow:
P1
cos 1
P1
3 1 2
1 1 2
a1 t  
a 2 t  
sin 
1  p̂1 a1 t  1
8 p̂1
4 p̂1

2 t   p̂ 2   2  
 p̂ 1 
1
P2
2
 p̂ 2 
(3)
cos  2
P2
3 2 2
1 2 2
sin 
a 2 t  
a1 t  
 2  p̂ 2 a 2 t  2
8 p̂ 2
4 p̂ 2
The full forms of constants  i ,  i ,  i and Pi could be
find using the proposed method. Here it was underlined
that these constants all rely on coefficients of coupling
properties via cofactors K 2si  , that  i depends of damping
coefficients of visco-elastic layer, ~i  and Pi depend of
excited amplitudes, and  i , i of non-linearity layer
properties. Coefficients i are coefficients of mode
mutual interactions. When we finally find the closed
forms of solutions then we are in position to make
conclusions about IS. If we denoted with  x1 , x1 ,... and
x2 , x 2 ,... the sets of dynamics variables which
corresponds to the states of first and second oscillators,
respectively, than the identical synchronization occurs
when, for sets of initial conditions  x1 0, x2 0,... and
x2 0, x 2 0,... , the difference of state variables
x2 t   x1 t   0 when time t   or in the form:
Figure 1. Periodically forced chain of two material particles
coupled with an element with damped, linear and nonlinear properties.
x2 t   x1 t   0
lim
t 
The system of the two coupled nonlinear differential
equations that corresponde to this model is:
This practically means that after enough period of time
the dynamics of oscillators becomes the same
xi 1 t   xi t  for i  1,2,..., n in proper approximation.
This equality in the unique state space presents subspace
(diagonal), and that equality represents convergence of
space vector xt  to that diagonal set x : xi 1  xi  , [4].
x1 t   2 1  x 2 t   x1 t   12  x2 t   x1 t   1 x2 t   x1 t 3  f 1 t 
x2 t   2 2  x 2 t   x1 t    22  x 2 t   x1 t    2 x 2 t   x1 t 3  f 2 t  (1)
where we use notation for the coefficients: of non-linear
coupling  i  c~ mi , of linear coupling i2  c mi , of
damped coupling. The functions of periodic excitation
are f i t   Fi mi  cosi t   i  . Proposed forms of the
solutions are:
xi t    as t K 2si cos s t 
(2)
With the unknown time depended amplitudes
ˆ
as t   Rs t e t and phase  s t    s t   s t  , where
i
are: K 2si co-factors of determinant corresponding to
basic homegenous coupled system ,  ˆi and p̂i are real
and imaginary parts of the appropriate pair of the roots
of the characteristic equation (see Ref. [10]).
Amplitudes a s t  and phases  s t  may be obtained by
using the Krilov-Bogolyubov-Mitropolyskiy asymptotic
method, if it was taken into account that defined task
satisfies all necessary conditions for applying
asymptotic method Krilov-Bogolyubov-Mitropolskiy
concerning small parameter  . The system of the first
order differential equations according to unknown
342 ▪ VOL. 42, No 4, 2014
.
(4)
While the treatment of just a two coupled elements can
be done within the framework of non-linear dynamical
systems, the description becomes much more
complicated for a large number of such elements.
The next model of interests is the model of chain lattice
composed of the four chains with eleven material
particles. The chains are orthogonally crossed on the
third and the ninth particles noted as a knots, so there is
two horizontal and two vertical chains, Fig. 2. Now we
have forty four coupled equations of motions in the
form:
3
3
mu uij  cu  ui 1. j  2ui . j  ui 1. j   c~u  ui 1. j  ui . j   c~u  ui . j  ui 1. j  
 bu  ui 1. j  2ui . j  ui 1. j   hUi , j cos ij t   ij 
for i  1..4, j  1....11
(5)
3. NUMERICAL SIMULATIONS OF IDENTICAL
SYNCHRONIZATION
To solve system of coupled differential equations (5)we
were forced to use numerical simulation to obtain
FME Transactions
conclusions about possibilities of knots IS. We assume
that horizontal and vertical directions of motion are
independent, so for a general trajectory of knots motions
we actually sum up the solutions from independent
directions. For possible identical synchronization of
vertical knots we are looking for the equality
x3 t   y3 t   u3 t   y9 t  or x9 t   v3 t   u9 t   v9 t  and for
possible
synchronization
x3 t   y3 t   x9 t   v3 t  or
of
the different forms of phase synchronization are
possible and the synchronization attractor is visible.
amplitude of excitation on main nod 33
A cos  33 t
3
2
4
6
7
8
5
5
6
6
7
xi
coefficient of linear stiffness
11
2
5
 33
3
1
4
2
6
7
8
11
11
n
4
4
4
coefficient of non linear stiffness
frequence of excitation
10
5
atack angle on knot 33
atack angle on knot 93




5

10
ui
 n2,3
3
2
1
t1
u3
D cos  n 2 , n  2t
C cos  n  2 ,3t
atack angle on knot 39
atack angle on knot 99
 n2,n2
5.......... ...
n
4
n
n
10
amplitude of excitation on main nod 33
u9
amplitude of excitation on main nod 93
10
amplitude of excitation on main nod 39
amplitude of excitation on main nod 99
5
2.08
2
Out[2]=
n
4
B cos  6 y t
10
5
5
10
5
10
u3
atack angle on knot 33
atack angle on knot 93

6y
coefficient of non linear stiffness
frequence of excitation
3
4

atack angle on knot 39
D cos6vt
5
atack angle on knot 99
 6v
t1

10
D cos 99t
ui
 99
9
10
6x
...

9
5
10
...
3
B cos 3, n  2t
 3,n2
3
.......... ...
coefficient of linear stiffness
A cos6 xt
3
1
A cos  33 t
2
1
vi
mi
3
0.45
1
4
8
 93
3
2
vi
mi
2
xi
10
7
C cos 93t
9
1
1
B cos39t
 39
9
4
8
1
2
damping coefficient
horizontal
c i , c~i , bi
yi
3
5
4
ui
yi
amplitude of excitation on main nod 99
knots
u3 t   y9 t   u9 t   v9 t  and
1
2
 33
3
1
vi
mi
2
xi
ci , ~
c i , bi
damping coefficient
ci , ~
c i , bi
1
10
amplitude of excitation on main nod 39
similar for the opposite combinations of knots. We also
can provide different direction and position of the
external excitation by changing the angle of attack on
the particles ij and also position of the forces itself,
Fig.2.
yi
u9
amplitude of excitation on main nod 93
10
11
C cos6ut
...
1
3
2
 6u
...
n
4
amplitude of excitation on main nod 33
u9
amplitude of excitation on main nod 93
n
n
10
amplitude of excitation on main nod 39
amplitude of excitation on main nod 99
Figure 2. Crossed chains with knot external excitation or
with excitation in the middle of the chains
In the left part of the Fig.2 all four knots are attacked
with external forces and we changed amplitudes and
angles of attack like as values of coupling coefficients
to synchronize them.The right part of the Fig. 2 presents
possible attacks on the middles of the vertical chains,
which perhaps give possibility of the diagonal
symmetry of the lattice with synchrony of the opposite
knots.
We
were
looking
for
the
diagonal x3 t   y3 t   u3 t   y9 t  for exploration of
synchronization of nodes on first vertical chain (nodes
33 and 39). That was naturally possible in the
completely symmetric lattice when external excitations
perform for notes in the same manner. The nodes adjust
the resulting movements and the synchronization
attractor is visible after transition changes, Fig. 3. But
for the value of the coefficient of linear coupling larger
than 2.08, Fig. 3, middle part, the diagonal attractor is
possible; the identical synchronisation is reached after
some initial period of time called synchronization time.
By increasing the damping coefficient the
synchronisation time is smaller, the nodes are
synchronised faster when the coupling is with greater
damping coefficient. The following observation was IS
of the opposite nodes 33 and 99, Fig.4, for the same
case of symmetrically excited lattice. Synchronization
of opposite nodes is less possible even though for larger
values of coefficient of linear coupling, upper part of the
Fig.4. But only increasing the damping coefficient the
IS emerges even for the less values of the coupling
strength, lower part of Fig. 4 the diagonal of
synchronisation is visible.
When lattice is not attacked symmetrically, for example
when external excitation acts on nodes 33, 39 and 99,
the IS is impossible, Fig.5. But after transition changes
FME Transactions
damping coefficient
5
1.405
coefficient of linear stiffness
Out[3]=
coefficient of non linear stiffness
frequence of excitation
10
5
u3
atack angle on knot 33
atack angle on knot 93
atack angle on knot 39
5
atack angle on knot 99
t1
10
Figure 3. Synchronization attractors for knots from left
vertical chain. External excitation acts on all four nodes.
amplitude of excitation on main nod 33
node99
amplitude of excitation on main nod 93
15
amplitude of excitation on main nod 39
ci , ~
c i , bi
yi
amplitude of excitation on main nod 99
1
xi
coefficient of linear stiffness
3
2
frequence of excitation
15
uatack angle on knot 33
n
5
4



10
5
C cos  n  2,3t
ui
atack angle on knot 39
 n2, 3
1
3
2



5
atack angle on knot 93
atack angle on knot 99
B cos 3, n  2t
 3, n2
3
.......... ...
4
4
coefficient of non linear stiffness
Out[5]=
1
2
 33
3
1
10
A cos  33 t
2
damping coefficient
vi
mi
5
10
15
node33
D cos  n  2, n  2t
 n  2 ,n  2
.......... ...
n
4
10
t1
n
n
15
amplitude of excitation on main nod 33
node99
amplitude of excitation on main nod 93
15
amplitude of excitation on main nod 39
amplitude of excitation on main nod 99
10
damping coefficient
1.215
5
coefficient of linear stiffness
Out[6]=
2.08
coefficient of non linear stiffness
15
10
5
5
10
15
node33
frequence of excitation
uatack angle on knot 33
5
atack angle on knot 93
atack angle on knot 39
10
atack angle on knot 99
t1
15
Figure 4. Synchronization attractors for opposite knots 33
and 99. External excitation acts on all four nodes.
The upper part of Fig.5 presents phase synchronisation
3:1, and the lower part phase synchronisation 1:1. By
increasing coefficient of linear coupling the possibilities
of synchronization increase after transition process,
middle part of the Fig.5. Increasing the frequency of
excitations, the phase locking 1:1 of nodes gets better
possibilities.
VOL. 42, No 4, 2014 ▪ 343
amplitude of excitation on main nod 33
1
amplitude of excitation on main nod 39
 33
3
2
1
1
A cos  33 t
3
damping coefficient
vi
10
mi
2
xi
amplitude of excitation on main nod 99
u9
ci , ~
c i , bi
yi
amplitude of excitation on main nod 93
2
3
.......... ...
n
4
4
period of time lost IS, it is necessary that signal from
external excitation from the middle of chain reachs to
the node after some initial period and to disturb the
synchrony. Increasing the time period we lost
synchrony, Fig. 6 lower part.
4
5
0.26
coefficient of linear stiffness


0.535
In[2]:=
coefficient of non linear stiffness



10

5
5
frequence of excitation
 93
3
2
1
amplitude of excitation on central nod 61h
n
4
6
amplitude of excitation on central nod 62h
5
ugao udara na cvor 93
u9
amplitude of excitation on central nod 61v
 99
.......... ...
ugao udara na cvor 33
u3
10
D cos  99t
C cos  93t
ui
0.17
amplitude of excitation on central nod 62v
n
n
4
damping coefficient
ugao udara na cvor 39
coefficient of linear stiffness
ugao udara na cvor 99
2
1.335
10
t1
In[12]:=
coefficient of non linear stiffness
frequence of excitation
6
4
2
2
4
u3
6
angle of attack on nod 61h
2
angle of attack on nod 61v
angle of attack on nod 62h
4
angle of attack on nod 62v
amplitude of excitation on main nod 33
u9
amplitude of excitation on main nod 93
t1
10
296
amplitude of excitation on main nod 39
6
amplitude of excitation on main nod 99
damping coefficient
5
Figure 7. Identical synchronization of nodes on first vertical
chain (nodes 33 and 39). External excitations act on middle
dots in vertical and horizontal chains
0.26
coefficient of linear stiffness
1.195
In[6]:=
coefficient of non linear stiffness
10
5
5
u3
10
frequence of excitation
0.138
ugao udara na cvor 33
5
ugao udara na cvor 93
ugao udara na cvor 39
ugao udara na cvor 99
10
t1
amplitude of excitation on main nod 33
u9
amplitude of excitation on main nod 93
10
amplitude of excitation on main nod 39
amplitude of excitation on main nod 99
damping coefficient
5
coefficient of linear stiffness
coefficient of non linear stiffness
In[8]:=
frequence of excitation
0.272
10
5
5
u3
10
ugao udara na cvor 33
ugao udara na cvor 93
ugao udara na cvor 39
5
ugao udara na cvor 99
t1
10
Figure 5. Synchronization attractors for knots from left
vertical chain. External excitation acts on nodes 33; 93 and
99.
amplitude of excitation on central nod 61h
u9
amplitude of excitation on central nod 61v
6
c i , c~i , bi
yi
amplitude of excitation on central nod 62h
1
amplitude of excitation on central nod 62v
2
xi
damping coefficient
1
3
n
4
4
4
2
coefficient of non linear stiffness
In[9]:=
2
6x
...
...
3
2
1
A cos 6 x t
4
3
coefficient of linear stiffness
vi
mi
By increasing coefficient of linear coupling the
period of time in a synchrony of nodes is longer, Fig.7,
we can notice that the period of time is the same like as
on the lower part of Fig.6, but there is diagonal line
representing IS because of the greater value of the
coefficient of the linear coupling.
When nodes from vertical chains share the signal from
the external excitation that acts on the middle point of
vertical chain, Fig. 8, then the initial identical
synchronization is possible with transition to phase
synchronization. If we retain all parameters on the same
values and only change the position of the external
excitations that acts on the middle points of the
horizontal chains, Fig. 9, then we find out identical
synchronisation of left vertical nodes. This is due to the
fact that excitation is symmetric in the latter case, and in
the first case the influence of the signal from the parallel
chains that has time delay causes the transition to phase
synchronization.
amplitude of excitation on central nod 61h
B cos6 y t
frequence of excitation

angle of attack on nod 61h
6
4

6 y
2

2

D cos6vt
u9
amplitude of excitation on central nod 61v
 6v
4
6
u3
6
amplitude of excitation on central nod 62h
ci , ~
c i , bi
yi
1
amplitude of excitation on central nod 62v
angle of attack on nod 61v
vi
mi
2
ui
angle of attack on nod 62v
 6u
...
...
1
t1
3
2
4
damping coefficient
C cos6ut
3
coefficient of linear stiffness
n
4
n
3
n
n
4
2
4
4
B cos  6 y t
frequence of excitation
ugao udara na cvor 61h
6
2
3
...
...
2
1
coefficient of non linear stiffness
In[13]:=
163.
xi

6
4
2

D cos 6 v t

6 y
2
 6v
4

u3
6
ugao udara na cvor 61v
ugao udara na cvor 62h
2
ui
...
ugao udara na cvor 62v
3
2
1
...
n
4
4
t1
n
amplitude of excitation on central nod 61h
1
4
2
angle of attack on nod 62h
n
u9
amplitude of excitation on central nod 61v
6
6
amplitude of excitation on central nod 62h
amplitude of excitation on central nod 62v
4
damping coefficient
coefficient of linear stiffness
2
coefficient of non linear stiffness
In[10]:=
frequence of excitation
angle of attack on nod 61h
6
4
2
2
4
6
u3
Figure 8. Identical synchronization of nodes on first vertical
chain (nodes 33 and 39). External excitations act on middle
dots in vertical chains
angle of attack on nod 61v
2
angle of attack on nod 62h
amplitude of excitation on central nod 61h
angle of attack on nod 62v
u9
amplitude of excitation on central nod 61v
4
t1
6
amplitude of excitation on central nod 62h
296
c i , c~i , bi
yi
1
vi
mi
amplitude of excitation on central nod 62v
4
2
6
damping coefficient
xi
coefficient of linear stiffness
3
1
3
2
Figure 6. Synchronization of nodes on first vertical chain
(nodes 33 and 39). External excitations act on middle dots
in vertical and horizontal chains
For the case when external excitations act on the
middle of the chains like it was presented on the right
part of Fig. 2, we have interesting phenomena of initial
synchronization of knots, the diagonal line that is visible
in the beginning of the motion, Fig. 6 upper part. In
initial periods nodes are synchronize but after some
344 ▪ VOL. 42, No 4, 2014
Out[13]=
frequence of excitation
ugao udara na cvor 61h
2
3
n
4
2
4
coefficient of non linear stiffness
1
A cos 6 xt
6x
...
...
4


6
4
2

2

4
6
u3
ugao udara na cvor 61v
ugao udara na cvor 62h
ugao udara na cvor 62v
2
ui
 6u
...
...
1
3
2
C cos 6u t
n
4
4
t1
n
n
6
Figure9. Identical synchronization of nodes on first vertical
chain (nodes 33 and 39). External excitations act on middle
dots in horizontal chains
FME Transactions
The presented numerical simulation was performed to
the identical synchronization mostly of the nodes from
the left vertical chain, but all the given conclusions have
relevance to synchronization of the other nodes.
4. CONCLUSION
The presented phenomenon of synchronization is very
useful for elucidate the overall dynamics of large
systems of material particles. The multi parametric
computations (in silico) experiment is very useful tool
for fast visual and parameter analysis of identical
synchronization threshold. In chains with non-linear
coupling the nonlinearity brings the properties of
exponentially divergence of trajectories of particles that
starts from very close points in phase space,
nevertheless herein was shown that even in that systems
one may find synchronization, which is rather surprising
detail.
The identical synchronisation of all four knots of the
lattice is possible if every node is synchronized with the
other three knots, furthermore it is possible to conclude
about synchronization of all knots on the base of the
three knots synchronization only if it the external
excitations act on the symmetric position in the lattice.
The possibilities for identical synchronisation are
generally greater if the coefficients of coupling have
greater values, but in some cases it is possible that we
have ragged synchronization with increasing some
parameters. On the base of these multi parametric
numerical simulations it is easy to find out the needed
and enough values of parameters for reaching IS, i.e. to
find out synchronization threshold that varies with
values of system parameter and with position and values
of the external excitations. As it is shown, by measuring
the synchronization times for different cases of
excitations positions, that these simulations could be
used for measure of signal processing in these models of
chain lattices.
ACKNOWLEDGMENT
This research was sponsored by the research grant of the
Serbian Ministry of Education, Science and
Technological Development under the number ON
174001
REFERENCES
[1] Pikovsky, A.; Rosemblum, M.; Kurths, J.:
Synchronization: A Universal Concept in Nonlinear
Sciences. Cambridge University Press, ISBN 0521-53352-X, 2001.
[2] Fradkov A.L., Pogromsky A.Yu.: Introduction to
control of oscillations and chaos. Singapore: World
Scientific Publishers, 1998.
[3] González-Miranda, J. M.: Synchronization and
Control of Chaos. An introduction for scientists and
FME Transactions
engineers. Imperial College Press. ISBN 1-86094488-4, 2004.
[4] Pecora, L.M., Carroll, T.L., Johnson, G.A., and
Mar, D.J.: Fundamentals of Synchronization in
Chaotic Systems, Concepts and Applications,
Chaos, vol. 7, no. 4, pp. 520-543, 1997.
[5] Balanov et al.: Synchronization: From Simple to
Complex. Springer-Verlag, 2009.
[6] Hedrih (Stevanović) K., Hedrih A.:Eigen modes of
the double DNA chain helix vibrations , Journal of
Theoretical and Applied Mechanics(JTAM),No. 1
(48),2010, p-p. 219-231, ISSN 1429-2955.
[7] Hedrih A., Hedrih(Stevanovic) K., BugarskiB.:
Oscillatory Spherical net model of Mouse
ZonaPellucida. Journal of Applied Mathematics
and bioinformatics. 2013, vol.3, no.4, 225-268.,
ISSN: 1792-6602, Scienpress Ltd, 2013.
[8] Hedrih A.: Modelling oscillations of Zona Pellucida
before and after fertilization. ENOC Young
Scientist Prize, Paper. EUROMECH Newsletter 40,
Dec. 2011, European Mechanics Society, 40, pp. 614.
[9] Simonović J.: Synchronization in Coupled Systems
with Different Type of Coupling Elements,
Differntial Equations and Dynamical Systems,
Volume 21, Issue 1 (2013), Pp. 141-148, ©
Springer 2013.
[10] Hedrih K., Simonović J.: Non-linear dynamics of
the sandwich double circular plate system,
International Journal of Non-Linear Mechanics 45
(9),2010, p-p: 902-918.
СИНХРОНИЗАЦИЈА У ЛАНЦИМА
МАТЕРИЈАЛНИХ ТАЧАКА СА
НЕЛИНЕАРНИМ СВОЈСТВИМА
Јулијана Д. Симоновић
Рад представља прелепе могућности идентичне
синхронизације у динамици системима ланаца
материјалних тачака. Спрежући елементи између
материјалних тачака имају својства пригушења,
линеарне и нелинеарне еластичнисти. Једна или
више материјалних тачака ланца је периодички
побуђена. Ефекат синхронизације је мање или више
изражен зависно од својстава спрежучих елемената.
Више параметарска анализа је извршена у фазном
простору излазних променљивих спрегнутих
материјалних тачака. Закључна разматрања садрже
напомене о природи спрежућих елемената и
симетрији у систему позиција дејства спољашње
побуде које су потребне и довољне да би се
достигао праг идентичне синхронизације.
VOL. 42, No 4, 2014 ▪ 345