Visualization of Scenarios for the Transition of Oscillations from
Harmonic to Chaotic for a Micropolar Kirchhoff-Love Cylindrical Meshed
Panel
E.Yu. Krylova1, I.V. Papkova2, O. A. Saltykova2,V.A. Krysko2
Kat.krylova@bk.ru|ikravzova@mail.ru|olga_a_saltykova@mail.ru| tak@sun.ru
1
Saratov State University, Saratov, Russia;
2
Yuri Gagarin State Technical University of Saratov, Saratov, Russia;
On the basis of the kinematic hypotheses of the Kirchhoff-Love built a mathematical model of micropolar cylindrical meshed panels
vibrations under the action of a normal distributed load. In order to take into account the size-dependent behavior, the panel material
is considered as a Cosser’s pseudocontinuum with constrained particle rotation. The mesh structure is taken into account by the
phenomenological continuum model of G. I. Pshenichnov. For a cylindrical panel consisting of two systems of mutually perpendicular
edges, a scenario of transition of oscillations from harmonic to chaotic is constructed. It is shown that in the study of the behavior of
cylindrical micropolar meshed panels it is necessary to study the nature of the oscillations of longitudinal waves.
Keywords:visualization of scenarios for the transition of oscillations into chaos, a mesh structure, a cylindrical panel, micropolar
theory, the Kirchhoff-Love model.
To account for the size-dependent behavior, a non-classical
1. Introduction continuum model based on the Cosser medium is considered,
To solve the problems of static and dynamic mesh plates, where, along with the usual stress field, torque stresses are also
panels and shells, mainly two computational models are used. It taken into account. This assumes that the displacement and
is a phenomenological continuum model and a discrete model. In rotation fields are not independent. In this case, the components
the continuum model, a mesh object consisting of a regular of the symmetric bending-torsion tensor are written as follows:
system of frequently located edges of one material is replaced by 2w 2w 1 2w 2w
an equivalent solid object having some averaged stiffness xx ; yy ; xy 2 2 ;
depending on the arrangement of the edges and their stiffness xy yx 2 y x
[1,3]. In the discrete model, the edges are represented by beam, 1 2 v 2u 1 2 v 2u
shell, or three-dimensional finite elements [2,5,7,11]. xz 2 ; yz .
Eachoftheseapproacheshasitsadvantages [4]. 4 x xy 4 yx y 2
Progress in micro-and nano-technologies leads to the interest We take the defining relations for the panel material in the
of scientists not only to the behavior of full-size mechanical 2
systems in the form of plates and shells [13,14,16], but also the form: mxx , mxу , mzx 1El xx , xу , zx ,
need to create mathematical models that take into account the
scale effects at the micro and nano level [10,12,19]. In most E E
xx exx eyy , x y, xy e , где ij - the
works on this subject linear models are used for numerical 1 2 1 xy
analysis [15,17,18,21,22]. However, there are experimental data
confirming the need to take into account the nonlinearity in components of the stress tensor, mij components of the moment
modeling the behavior of the objects under consideration [20]. tensor of higher order, E - Young’s modulus, - Poisson’s
Despite the large number of works devoted to the size-
dependent behavior of mechanical objects in the form of plates, ratio.
panels and shells, studies of the behavior of mesh plates and The equations of motion of a smooth plate element
shells based on theories that take into account the effects of scale equivalent to a mesh one, boundary and initial conditions are
is very small [6,8,9]. obtained from the Hamilton – Ostrogradsky energy principle:
N xx T 1 Yyz 1 2Yxz 2u
2
2. Problemstatement h 2 ,
x y 2 y 2
2 xy t
A mathematical model of oscillations of a micro-polar
N yy T 1 2Yxz 1 Yyz
2
2v
flexible rectangular cylindrical panel under the action of a h 2 ,
3 y x 2 x 2
2 xy t
transverse distributed pressure occupying a region in space R (1)
h h N xx w 2 w N yy w 2w 2H T w
area 0 x c;0 y b; z is constructed. N xx 2 N yy 2 2 2
2 2 x x x y y y xy x y
The panel consists n of sets of densely arranged edges of the
T w 2 w 2 M xx M yy 2H
2
same material, which allows the use of a phenomenological 2 4T 2 k y N yy
continuum model. Taking into account the Kirchhoff-love y x xy x 2
y 2
xy
2Yxx Yyy Yxy Yxy 2w w
2 2 2
hypotheses, the strain tensor components are written as:
2q h 2 h .
u 1 w w
2
v 1 w xy yx x y t t
2
w
2 2 2 2
exx z 2 ; eyy kyw z 2 ;
x 2 x x y 2 x y
Boundary conditions:
1 u v w w 2w
exy z .
2 y x x y xy
Where u , v, w - axial displacements of the middle surface of the
plate in the directions x, y , z respectively, k y - geometric
curvature parameter.
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
w M x w Yxy H Yxx Yyy Yxy N xxs A40 N xx A22 N yy A31T ;
N xx 2T 2
x x y x y y y x n N yys A22 N xx A04 N yy A13T ;
x
w M y w H Yxx Yyy Yxy T s A31 N xx A13 N yy A22T ;
N yy 2T 2 0
y y x x x x y n M xxs A40 M xx A22 M yy A31H ;
y
M Y 2H Y Y 0;
xx xy n
x
xx yy n
y
M yy A22 M xx A04 M yy A13 H ;
2H Y Y M Y 0;
xx yy n yy xy n
H s A31M xx A13 M yy A22 H ; (2)
Y A40Yxx A22Yyy A31Yxy ;
x y s
1 Yyz 1 Yxz
xx
1 Yxz
N xx T 0; Yyys A22Yxx A04Yyy A13Yxy ;
2 y nx 2 y 2 x n
y
Yxys A31Yxx A13Yyy A22Yxy ;
1 Yxz 1 Yyz 1 Yyz 1
N xy N yy 0; Yxz 0; Yyzs A11Yxz A02Yyz A01Yzz ;
2 x 2 y n 2 x n 2 n y
x y
Yxzs A20Yxz A11Yyz A10Yzz .
Yxz nx Yyz ny 0; 1 Yxz 1 Yyz 0; 1 Yyz 0.
2 nx
Substituting expressions (2) into equations (1), we obtain a
2 nx 2 ny
resolving system of equations of motion for a smooth micropolar
Here the expression for the classical force and torque: cylindrical Kirchhoff-Love panel equivalent to the original mesh
h
2
panel.
N , N ,T , , dz,
xx yy xx yy xy
In this model, the rigidity of the rods to bend in a plane
h tangent to the middle surface of the panel is not taken into
2
account, so the orders of systems of differential equations
h
2 describing the behavior of grid and solid panels coincide. At the
M , M , H , , zdz,
xx yy xx yy xy same time, the formulations of the boundary conditions of the
h
corresponding boundary value problems coincide.
2
The scenario of transition of oscillations from harmonic to
as well as expressions for the forces caused by instantaneous
chaotic cylindrical panel with two families of edges is
stresses:
h h h investigated 1 45o ,2 135o , 1 2 , a1 a2 a (Fig
2 2 2
Yxx mxx dz , Yxy mxy dz , Yxz mxz dz , h2
x y. .1). Taking into account dimensionless parameters: u u,
h
h
h c
2 2 2
h2 h Eh 4 1 E
The stresses arising in the equivalent smooth panel v v , k y 2 k y , q 2 2 q ;. t b t, ,
associated with the stresses in the edges that make up the angles b b cb E b
j with the abscissa axis will have the form: x cx , y b y , w hw , h , a ha , l hl , where -
n xj j Cos j Sin j n xj j Cos 2 j dissipation factor, - the density of the panel material,
xy , xx
q q0 Sin pt - external normal load, q0 and p - its intensity
,
j 1 aj j 1 aj
n xj j Sin 2 j n mxj j Cos j Sin j and frequency, t - time. The equations of motion of the element
yy , mxy , of the considered micropolar mesh cylindrical panel will take the
j 1 aj j 1 aj form (the line above the dimensionless variables is omitted):
n mxj j Cos 2 j n mxj j Sin 2 j
mxx , myy ,
j 1 aj j 1 aj
n mzxj j Cos j n mzxj j Sin j
mxz , myz , where a j -
j 1 aj j 1 aj
distance between edges of j-th family, j – the thickness of the
ribs, voltage index j are rods. The physical relations for the mesh
plate are determined based on the Lagrange multiplier method:
xj xx Cos 2 j yy Sin 2 j xy Cos j Sin j ;
j xz Cos j yz Sin j ;
mxj mxx Cos 2 j m yy Sin 2 j mxy Cos j Sin j ;
mzj mxz Cos j m yz Sin j .
Fig.1.Panel mesh geometry.
n j Cos s j Sin k j
Subject to designation: Ask ; s, k 0,4
j 1 aj
expressions for classical forces and moments, as well as the
forces caused by the moment stresses of the cylindrical mesh
panel will take the form: (the upper index shows the account of
the mesh structure):
h 2 4u h 2 4 v h 2 4u h 2 4v 3. Scenarios of transition of oscillations of a
l 2 1 2 4 2 2 2 2 2 3 cylindrical panel to chaos
b y b xy c x y c x y
3
u2
v
2
b u
2 2 To visualize the scenarios of transition of oscillations of a
2 1 2 2 3 4 1 2 2 micropolar mesh cylindrical panel from harmonic to chaotic for
y xy c x
deflection and displacement, the following characteristics were
b w 2 w w w 2 w
4k y 1 4 1 2 8 constructed and analyzed: signal, Fourier spectrum, wavelet 2D
h x y x y xy and 3D spectra based on the mother wavelet Morle, phase and
b 2 w 2 w 8a 1 h 2 2u
2 modal portraits, signs of largestLyapunov exponents.
4 1 2 The following is a scenario of transition of oscillations of a
c x x 2 c 2 t 2 grid cylindrical micropolar panel from harmonic to chaotic
h 2 4u h 2 4v h 2 4u h 2 4v (Table 1-3).
l 2 1 2 2 2 2 2 3 2 4 The parameters of the experiment: l 0.5 , c b 1 ,
b xy b x y c x y c x
3
h 0.2 , 1 , a 0.2 , 0.3 , p 5 , t [0;512] ,
c w
2
c v
2 2
2u
4k y 1 4 1 2 2 2 3 q0 [0;200] .From the data collected in the tables it can be seen
bh y b y xy
that in addition to the characteristics of the deflection, the nature
2v c 2 w 2 w w 2 w
2 1 4 1 2 8 of the oscillations of the longitudinal waves should be studied,
x 2
b y y 2
x yx which will allow a more accurate picture of the nature of the
w 2 w 8a 1 h 2 2v oscillations of the system. At load amplitude 𝑞0 = 0.1, the
2
4 1 Fourier power spectrum for the deflection shows harmonic
y x 2 b 2 t 2 oscillations, but the Lyapunov exponent for the deflection is
c w 2 4
c w 2 4
1 6l 1 bh 4 1 l 1
positive. This discrepancy is explained by the fact that the signal
2 2
of the displacement function u has a chaotic component at low
y 4
b x y 2 2 2
frequencies. Harmonics at the same frequencies are present in the
b2 4w c2
1 6l 2 1 2 4 12k y2 1 2 w deflection signal, but the Fourier spectrum does not display them.
c x b These frequencies demonstrate the wavelet spectrum, so it is
2
c 2 v b 2 u c 2 w necessary to consider the Fourier spectrum and the wavelet
12k y 1 12k y 1 2 6k y 1 spectrum together. As the load increased, a harmonic appeared
bh y h x bh y
in the signal at an independent frequency 𝜔1 . When the
c 2 w 2v c2 2w c 2 v 2 w amplitude of the load 𝑞0 = 190 phase portrait of the deflection
12 1 12k y 1 w 2 12 1 2
b y y
2 2
bh y b y y 2 shows chaotic oscillations and the power spectrum of the Fourier
2
transform of the oscillations at two frequencies. Thus, to
w 2 w 2 w u 2u w
18 1 12 1 2 12 1 2 determine the type of deflection oscillations, it is also necessary
y y y x y x
2
to consider the function of moving by 0x or 0y.
b w 2 w w w 2u
2 2
6k y 1 6 5 3 2 24 Table1. The characteristics of the deflection function w and
h x y x y xy the displacement function u
w 2v u 2 w v 2 w 𝑞0 = 0.1, 𝜔𝑝 = 5
24 24 1 24 1 Fourierspectrum Phaseportrait
x xy y xy x xy
w(0.5;0.5,t)
w w 2 w b 2 w 2u
24 3 5 12 1 2
-3
4
x 10
y x xy c x x 2
p
0
2
b w 2v b 2w c 2 v 2 w
12 1 12 k 1 w 12 1
w' t
S
c y x 2 h x 2 b 2 y x 2
y -5 0
2 -2
w 2 w b 2 w 2 w
2
6 3 5 18 1 2 -10
1 2 3 4 5 -4
y x c x x 2
2
-5 0
-4
5
w
u 2 w 24a 1 2 w
x 10
2
w u(0.5;0.5,t)
12 1 2 2q
x x 2
-11
t t
x 10
-8 1
Boundary conditions – rigid sealing at all ends of the panel: -10 0.5
u u v v
t
u v w 0, 0, 0, 0, 0, .
S
w'
-12 0
x y x y -0.5
-14
w w
0, 0 при x 1, y 1. -16 -1
x y 1 2 3 4 5 -10 -5 0 5
-11
w x 10
Initial conditions – zero. Table2. The characteristics of the deflection function w and
The nonlinear partial differential problem in spatial the displacement function u
coordinates is reduced to an ordinary differential problem by the
𝑞0 = 130, 𝜔𝑝 = 5
finite difference method with the second-order approximation of
accuracy. To do this, the derivatives of spatial variables are Fourierspectrum Phaseportrait
replaced by finite central differences. The Cauchy time problem w(0.5;0.5,t)
is solved by the Runge-Kutta method of the fourth order of
accuracy.
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Chaotic dynamics of flexible beams driven by external white
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