NOVEL PROCEDURE TO COMPUTE A CONTACT
ZONE MAGNITUDE OF VIBRATIONS OF
TWO-LAYERED UNCOUPLED PLATES
J. AWREJCEWICZ, V. A. KRYSKO, AND O. OVSIANNIKOVA
Received 15 February 2004
A novel iteration procedure for dynamical problems, where in each time step, a contacting plates’ zone is improved, is proposed. Therefore, a zone and magnitude of a contact
load are also improved. Investigations of boundary conditions’ influence on externally
driven vibrations of uncoupled two-layer plates, where for each of the layers, the Kirchhoff hypothesis holds, are carried out.
1. Introduction
Uncoupled multilayer plates create a complex dynamical structure, where depending on
input parameters and initial and boundary conditions, various vibration types appear.
In spite of undoubtable achievements in static and dynamical problems of nonlinear
theory of plates, the problems of contacts exhibited by multilayer plates subject to both
longitudinal and transversal time-changeable loads are less investigated.
2. Mathematical models
In this work, a model of two-layer construction composed of thin elastic rectangular
plates is studied [7]. The mean surface of an upper plate lies in plane z = 0, whereas
the mean surface of the lower one lies in plane z = (1/2)(δ1 + δ2 ) + h1 , where h1 is the
distance between plates, δ1 , δ2 are the thicknesses of upper and lower plates, correspondingly. The plates are contacting with each other through external surfaces, projected into
a corresponding mean surface, within a general Winkler’s hypothesis [5]. Note that each
of the plates is embedded in 3D space in R3 with attached coordinates. Namely, senses
of the axes Ox and Oy go into direction of mean surface of upper plate, whereas axis Oz
goes into Earth centre. In the given coordinates, the 3D plates spaces Ω read
δ1
δ1
≤z≤
,
2
2
1
1
Ω2 = x, y,z/(x, y) ∈ [0,a] × [0,b], δ1 − δ2 + h1 ≤ z ≤ δ1 + δ2 + h1 ,
2
2
Ω1 = x, y,z/(x, y) ∈ [0,a] × [0,b], −
(2.1)
where the space [0, a] × [0,b] defines rectangular plate shape; D1 , D2 are cylindrical plates
Copyright © 2005 Hindawi Publishing Corporation
Mathematical Problems in Engineering 2005:4 (2005) 425–435
DOI: 10.1155/MPE.2005.425
426
Novel procedure to compute a contact zone magnitude
stiffness. The governing equations are
E
E
D1 ∆2 w1 (x, y,t) + k w1 (x, y,t)Ψ(x, y,t) = q1∗ + k w2 + h1 Ψ(x, y,t),
h
h
E
E
2
∗
D2 ∆ w2 (x, y,t) + k w2 (x, y,t)Ψ(x, y,t) = q2 + k w1 + h1 Ψ(x, y,t),
h
h
(2.2)
where the function
1
1 + sign w1 − w2 − h1 ,
(2.3)
2
and Ψ = Ψ(x, y) plays a role of shells’ contact space Ω∗ indicator. Notice that if the initial
plates location (clearance function) and the loads do not lead to a contact between plates
during their deformations, then Ψ ≡ 0 and each of the plates vibrates independently. Otherwise, the governing equations are coupled.
The system (2.2) is associated with one of the following boundary conditions [6] on
the boundary ∂Ω1 ,
Ψ=
wi wi ∂Ωi
=
=
∂Ωi
∂2 wi ∂wi = 0,
∂ni ∂Ωi
= 0,
∂ni ∂Ωi
(i = 1,2).
(2.4)
(2.5)
Owing to D’Alembert principle, qi∗ include both inertial and damping forces acting
on the ith plate of the form
qi∗ (x, y,t) = qi (x, y,t) −
γi ∂2 wi
∂wi
− εi
.
g ∂t 2
∂t
(2.6)
The system (2.2) is of high order with respect to time and spatial coordinates (x, y).
The computational process is as follows. In each time step, the following values are ac•
•
counted from a previous step wi |tk−1 = wi (x, y,tk−1 ); wi |tk−1 = wi (x, y,tk−1 ), and in order
to improve a contact zone, the following iteration procedure for time tk is applied:
kE
kE Ψw1(m+1) = q1∗n + Ψ w2(m) + h1 ,
h1
h1
kE
kE D2 ∆2 w2(m+1) + Ψw2(m+1) = q2∗n + Ψ w1(m) + h1 .
h1
h1
D1 ∆2 w1(m+1) +
(2.7)
First PDEs (2.2) are reduced to the Cauchy problem through the second-order method
of finite difference, then the problem is solved using fourth-order Runge-Kutta method.
In each time step, the Gauss iteration procedure (2.7) is carried out, and the system order
is reduced twice, which is important owing to computation time (see also [3]). Finishing
•
the iteration procedure (2.7), the obtained values of wi and wi serve as an initial condition for a next step of the Runge-Kutta method. The similar like methods have been also
applied in [1, 2, 4]. The mentioned procedure allows for a contact zone improvement.
Owing to the Runge principle, it has been found that the optimal step with respect to
spatial coordinates is defined through a partition of space Ω j into 15 × 15 parts, whereas
J. Awrejcewicz et al. 427
time step is equal to ∆t = 1 · 10−3 . In what follows, vibrations of two-layer plates with various boundary conditions along their contours are studied. The following four variants
of the boundary conditions are accounted:
(i) two plates are clamped along their contours (boundary conditions (2.4));
(ii) two plates are ball-type supported (boundary conditions (2.5));
(iii) upper plate is clamped along its contour (boundary conditions (2.4)), whereas
lower plate is ball-type supported along its contour (boundary conditions (2.5));
(iv) upper plate is ball-type supported along its contour (boundary conditions (2.5)),
whereas lower one is clamped (boundary conditions (2.4)).
3. Two plates are clamped along their contours (boundary conditions (2.4))
Assume that two plates have same thickness (δ1 = δ2 = δ), and the frequency of excitation ω0 = ωP = 9.973, where ω0 is the frequency of a linear vibration of one-layer plate,
ε1 = ε2 = 0.6. The clearance between plates is h1 = h1 /δ = 0.007 (h1 is the nondimensional parameter). Both plates are subject to sinusoidal load of the form qi = Qi sinωpt.
Boundary conditions (2.4) are applied, and the initial conditions read
•
w1 t=0 = w1 t=0 = 0.
(3.1)
Recall that in each time step, the iteration procedure (2.7) is applied. In Table 3.1, parts
•
of time histories wi (0,5;0,5;t), 26 ≤ t ≤ 32, phase portraits wi (wi ), and power spectrum
s(ω) are presented only for lower plate, since the vibrations of lower plate are mirror reflections of the upper plate. Furthermore, the results are only given for the plate centre,
since vibrations of other points are synchronized with those of the centre. During plates
contact, there exist time instants where vibrations process is an unstationary one, then after a long time, it achieves a stationary state. For q1 = −q2 < 0.5, the plates do not contact
with each other, and harmonic vibrations occur.
This observation is confirmed through the following characteristics. In power spectrum, only one frequency is visible, and an ellipse occurs in the phase portrait. However,
for q1 = −q2 = 2 (a contact between plates occurred), a picture is changed: vibrations
are no longer harmonic, and the phase portrait is of more complexity. Increasing the
transversal load, a Hopf bifurcation occurs with a period tripling (q1 = −q2 = 11) (see
[8]). Further, an interlace of vibrations on the excitation frequency and on the frequencies of period tripling and number eight is observed.
4. Boundary conditions (2.5)
Consider two-layer plate type construction with the same parameters as in the previously
analyzed case, but with boundary conditions (2.5). Similarly as in the previous case, for
q1 = −q2 = 0.5, harmonic vibrations occur. However, now bifurcations appear already for
q1 = −q2 = 1.5. Increasing qi , a picture of plates bending becomes more complicated. For
q1 = −q2 = 3, period tripling occurs, then a transition to chaos takes place (q1 = −q2 =
4). This is clearly expressed in both phase portrait and power spectrum. For q1 = −q2 = 5,
again Hopf bifurcation appears, then its collapse takes place (q1 = −q2 = 6). The scenario
is composed of interlace of Hopf bifurcations and a transition into harmonic vibrations.
428
Novel procedure to compute a contact zone magnitude
Table 3.1
Q
Signal
w(0.5, 0.5)
Power spectrum
×10−415
×10−485
0.5
Phase portrait
80
75
70
65
60
55
26 27 28 29 30 31 32
t
10
5
W
0
−5
−10
−15
−15
−5
5
W t
(a)
15
×10−3
10−12
10−14
S 10−16
10−18
10−20
10−22
w(0.5, 0.5)
2
(b)
26 27 28 29 30 31 32
t
2
W 0
−2
−4
−6
10−19
−4
0
W t
4
10−23
8
×10−2
w(0.5, 0.5)
W
w(0.5, 0.5)
(j)
w(0.5, 0.5)
−0.1
0.1
0.3
W t
26 27 28 29 30 31 32
t
(m)
ω
10−9
S
10−13
10−17
−0.1
0.1
0.3
10−21
W t
0
1
0
−1
−2
−3
1
1.5
(l)
−0.3 −0.1
0.1
W t
(n)
0.5
ω
(k)
W
1.5
(i)
×10−2 2
×10−2 2
1
10−6
ω
10−10
1/3ω
S 10−14
2/3ω
10−18 1/6ω
1/2ω
5/6ω
10−22
0
0.5
1
1.5
ω
(h)
1
W 0
−1
−2
−0.3
0.5
ω
×10−2 2
26 27 28 29 30 31 32
t
0
(f)
1
0
−1
−2
−0.3
(g)
1
0
−1
−2
1/2ω
(e)
×10−2 2
×10−2 2
ω
S 10−15
−8
26 27 28 29 30 31 32
t
14
1.5
10−7
1
0
−1
−2
1
0
−1
−2
−3
1
10−11
×10−2 2
12
0.5
(c)
×10−34
(d)
11
0
ω
×10−3 4
2
0
−2
−4
−6
ω
0.3
10−9
10−11
10−13
S 10−15
10−17
10−19
10−21
ω
1/3ω
2/3ω
0
0.5
1
ω
(o)
1.5
J. Awrejcewicz et al. 429
Table 3.1. Continued.
Q
Signal
Phase portrait
w(0.5, 0.5)
×10−2 3
18
×10−2 3
2
1
0
−1
−2
−3
26 27 28 29 30 31 32
t
2
1
W 0
−1
−2
−3
−0.4 −0.2
(p)
w(0.5, 0.5)
28
26 27 28 29 30 31 32
t
2
W 0
−2
−4
−6
−0.6
w(0.5, 0.5)
0
0.5
1
1.5
ω
(r)
26 27 28 29 30 31 32
t
0.2
0.4
ω
10−8
S
10−12
1/7ω
10−16
−0.2
0.2
10−20
0.4
W t
2
W 0
−2
−4
−6
−8
−4
0
W t
2/7ω
0
0.6
0.8
1
(y) w(0.5, 0.5) = A for q1 = −q2 = 11, 29
contact points
5/7ω
0.5
1
1.5
(u)
4
8
10−6
10−10
S 10−14
10−18
10−22
×10−2
ω
0
0.5
1
1.5
ω
(w)
5
4
3
2
1
0
−1
3/7ω
6/7ω
4/7ω
ω
×10−34
(v)
5
4
3
2
1
0
−1
0
(t)
×10−2 6
32
0.4
ω
×10−2 4
(s)
4
2
0
−2
−4
−6
0 0.2
W t
10−9
10−11
10−13
S 10−15
10−17
10−19
10−21
(q)
×10−2 4
2
0
−2
−4
−6
Power spectrum
(x)
0
0.2
0.4
0.6
0.8
1
(z) w(0.5, 0.5) = B for q1 = −q2 = 28, 56
contact points
5. Vibrations associated with different boundary conditions
In Table 5.1, same characteristics as in Table 5.2 are reported, but now the upper layer is
ball-type supported along the contour (boundary conditions (2.5)), whereas the lower
layer is clamped (boundary conditions (2.4)).
For q1 = −q2 = 0.5, vibrations of two plates are harmonic, and each of the plates vibrates with its own frequency. For q1 = −q2 = 1.5, when a contact between plates occurs,
430
Novel procedure to compute a contact zone magnitude
Table 5.1
Q
Signal
0.5
w(0.5, 0.5)
×10−3 3
2
1
0
−1
−2
−3
Phase portrait
×10−3 3
26 27 28 29 30 31 32
t
2
1
W 0
−1
−2
−3
−15 −5
(a)
w(0.5, 0.5)
1.5
W
2
0
−2
−4
−6
w(0.5, 0.5)
w(0.5, 0.5)
ω
−2
2
W t
1/2ω
10−14
10−18
10−22
×10−2
0
(e)
W
26 27 28 29 30 31 32
t
0.2 0.4 0.6 0.8
ω
1
(f)
2
0
−2
−4
−6
−8
−10 −6 −2
2
6
10−8
10−12
S 10−16
10−20
10−24
×10−2
W t
ω
1/3ω
0
(h)
2/3ω
0.2 0.4 0.6 0.8
ω
1
(i)
×10−3 6
4
2
0
−2
−4
−6
−8
26 27 28 29 30 31 32
t
4
2
0
W −2
−4
−6
−8
−15
(j)
−5
W t
5
×10−2
10−7
ω
3/4ω
10−11 1/4ω 1/2ω
−
15
S 10
10−19
10−23
0 0.2 0.4 0.6 0.8 1
ω
(k)
(l)
×10−3 6
×10−3 6
w(0.5, 0.5)
1
×10−34
×10−3 6
5
0.2 0.4 0.6 0.8
ω
(c)
S
(g)
4
0
10−10
−6
26 27 28 29 30 31 32
t
×10−3 4
3
15
×10−3
ω
(b)
(d)
2
0
−2
−4
−6
−8
5
W t
10−8
10
10−
10−12
S 10−14
16
10−
10−18
10−20
×10−34
×10−34
2
0
−2
−4
−6
Power spectrum
4
2
0
W −2
−4
−6
−8
−10
2
−2
−6
−10
26 27 28 29 30 31 32
t
(m)
−10 −5 0
W t
(n)
5 10 −2
×10
10−8
10−12
S 10−16
10−20
10−24
1/2ω
0
0.2 0.4 0.6 0.8
ω
(o)
ω
1
J. Awrejcewicz et al. 431
Table 5.1. Continued.
Q
Signal
Phase portrait
×10−3 6
×10−3 6
w(0.5, 0.5)
4
2
0
−2
−4
−6
−8
−10
6
2
10−9
10−13
W −2
−10
26 27 28 29 30 31 32
t
w(0.5, 0.5)
10
ω
S 10−17
10−21
−6
−10 −5 0
10−25
5 10
×10−2
W t
(p)
×10−3
Power spectrum
0
0.2 0.4 0.6 0.8
ω
(q)
5
0
×10−35
−5
W −5
−10
−10
(r)
10−9
10−11
10−13
−15
S 10−17
10
10−19
10−21
10−23
0
−15
−15
26 27 28 29 30 31 32
t
−3
−1
1
W t
(s)
1
×10−2
ω
1/2ω
0
0.2 0.4 0.6 0.8
ω
(t)
1
(u)
×10−3 10
5
0
−5
W −10
−15
−20
−25
20
w(0.5, 0.5)
×10−3 10
5
0
−5
−10
−15
−20
−25
−0.4
26 27 28 29 30 31 32
t
(v)
5
4
3
2
1
0
−1
0
0.2
0.4
0
W t
10−9
10−11
10−13
−15
S 10−17
10
10−19
10−21
10−23
0.4
0
0.2 0.4 0.6 0.8
ω
(w)
0.6
0.8
1
(y) w(0.5, 0.5) = C for q1 = −q2 = 4, 21
contact points
ω
1/2ω
(x)
2
1.5
1
0.5
0
−0.5
−1
0
0.2
0.4
0.6
0.8
1
(z) w(0.5, 0.5) = D for q1 = −q2 = 6, 56
contact points
1
432
Novel procedure to compute a contact zone magnitude
Table 5.2
Signal
×10−3 3
2
1
0
−1
−2
−3
w(0.5, 0.5)
Q = 0.5 ω = 5.7
Q
Phase portrait
×10−3 3
26 27 28 29 30 31 32
t
2
1
W 0
−1
−2
−3
−15 −5
(a)
w(0.5, 0.5)
ω = 9.973
80
75
70
65
60
55
26 27 28 29 30 31 32
t
×10−3 8
w(0.5, 0.5)
Q = 1.5 ω = 5.7
0
0.5
1
1.5
ω
(c)
2
1
W 0
−1
−2
−3
−15 −5
5
W t
15
×10−3
10−12
10−14
−16
S 10−18
10
10−20
10−22
ω
0
0.5
1
1.5
ω
(e)
(f)
×10−3 8
4
4
0
W
−4
−8
0
−4
−8
−8 −4
26 27 28 29 30 31 32
t
(g)
×10−3 12
w(0.5, 0.5)
15
×10−3
ω
(b)
(d)
ω = 9.973
5
W t
10−8
10−10
10−12
14
S 10−
10−16
10−18
10−20
×10−3 3
×10−4 85
0
W t
4
×10−2
ω
10
10−
1/2ω
10−12
−
14
10
S 10−16
3/4ω
10−18 1/4ω
10−20
−
22
10
0
0.5
1
ω
(h)
1.5
(i)
×10−3 12
10
8
6
4
2
Power spectrum
10
W
26 27 28 29 30 31 32
t
(j)
10−10
8
6
S
4
10−18
2
10−22
−6
−2
2
W t
(k)
6
×10−2
1/7ω
10−14
2/7ω 4/7ω 6/7ωω
3/7ω
5/7ω
0
0.5
1
1.5
ω
(l)
in upper-plate power spectrum, two Hopf bifurcations are remarkable, whereas period
seven is associated with lower plate (see the power spectrum). For q1 = −q2 = 3, in the
power spectrum of lower plate, also two Hopf bifurcations appear, and two plates begin
to vibrate with one frequency.
In Table 5.3, the same characteristics as in Table 5.1 are reported, but for different
boundary conditions. Namely, upper plate is clamped through its contour (boundary
J. Awrejcewicz et al. 433
Table 5.2. Continued.
Q
Signal
Phase portrait
w(0.5, 0.5)
Q = 3 ω = 5.7
10
5
0
−5
−10
−15
12
W
0
−15
w(0.5, 0.5)
5
W t
10−23
×10−2
0
0.5
1
1.5
ω
(o)
×10−310
12
5
0
W
−5
−10
−15
8
4
0
−15
26 27 28 29 30 31 32
t
(p)
S
−5
5
W t
4
2
0
−2
−4
−6
3/4ω ω
10−10 1/4ω
10−14
1/2ω
10−18
10−22
×10−2
0
0.5
1
1.5
ω
(q)
×10−2
w(0.5, 0.5)
−5
3/4ω
(n)
×10−3 16
ω = 9.973
−15
S 10
10−19
4
26 27 28 29 30 31 32
t
ω
10−11 1/4ω 1/2ω
8
(m)
Q = 24 ω = 5.7
Power spectrum
×10−3 16
×10−3
(r)
×10−2 8
6
4
W 2
0
−2
−4
−6
26 27 28 29 30 31 32
t
3/4ω ω
10−9 1/4ω
−
13
S 10
10−17
1/2ω
−2
(s)
2
W t
10−21
6
×10−2
0
0.5
1
1.5
ω
(t)
(u)
w(0.5, 0.5)
ω = 9.973
×10−2 6
×10−2 4
4
2
0
W
−2
−4
2
0
−2
−6
26 27 28 29 30 31 32
t
(v)
8
4
0
−4
0
0.2
0.4
S
10−13
10−17
−8 −4
0 4
W t
10−21
8
×10−2
1/2ω
0
0.5
0.6
0.8
1
(y) w(0.5, 0.5) = E for q1 = −q2 = 24, 104
contact points
1
1.5
ω
(w)
12
3/4ω ω
10−9 1/4ω
(x)
4
3
2
1
0
−1
0
0.2
0.4
0.6
0.8
1
(z) w(0.5, 0.5) = F for q1 = −q2 = 24, 36
contact points
434
Novel procedure to compute a contact zone magnitude
Table 5.3
Signal
Phase portrait
80
75
70
65
60
55
26 27 28 29 30 31 32
t
2
1
W 0
−1
−2
−3
−15 −5
×10−3 3
2
1
0
−1
−2
−3
2
1
W 0
−1
−2
−3
−15 −5
26 27 28 29 30 31 32
t
5
0
−5
−10
−15
26 27 28 29 30 31 32
t
5
W t
−20 −10
1.5
ω
10−12
10−16
10−20
15
×10−3
0
0.5
1
1.5
ω
(f)
×10−310
5
W 0
−5
−10
−15
1
(c)
0
W t
10
10−9
3/4ω ω
10−11 1/4ω
10−13
S 10−15
10−17
1/2ω
10−19
−
21
10
0
0.5
1
ω
×10−2
(h)
1.5
(i)
×10−3 30
×10−3 25
w(0.5, 0.5)
0.5
(e)
×10−3 10
w(0.5, 0.5)
Q = 6 ω = 9.973
0
ω
S
(g)
ω = 5.7
15
×10−3
ω
10−8
(d)
20
S
0
26 27 28 29 30 31 32
t
3/4ω ω
10−9 1/4ω
W 10
(j)
5
4
3
2
1
0
−1
5
W t
10−12
10−14
10−16
S
10−18
10−20
10−22
(b)
×10−3 3
w(0.5, 0.5)
ω = 5.7
(a)
20
15
10
5
0
−5
−10
Power spectrum
×10−3 3
×10−4 85
w(0.5, 0.5)
Q = 0.5 ω = 9.973
Q
10−13
1/2ω
10−17
−10
−0.2
0
W t
10−21
0.2
0
0.5
1
1.5
ω
(k)
(l)
1.5
0
0.2
0.4
0.6
0.8
1
(m) w(0.5, 0.5) = K for q1 = −q2 = 6, 61
contact points
1
0.5
0
−0.5
−1
0
0.2
0.4
0.6
0.8
1
(n) w(0.5, 0.5) = M for q1 = −q2 = 6, 40
contact points
J. Awrejcewicz et al. 435
conditions (2.4)), whereas the lower plate is supported through the boundary conditions
(2.5).
For q1 = −q2 = 0.5, both plates vibrations are harmonic. For q1 = −q2 = 6, vibrations
of both plates are synchronized with one frequency. Then two Hopf bifurcations follow.
6. Conclusions
The carried out analysis exhibits complex vibrations of two-layer system of plates: series
of Hopf bifurcations occurs, where periods three, five, and seven Hopf bifurcations are
exhibited. The detected bifurcations in our complex system have been theoretically predicted by Sharkovskiy while analyzing the logistic curves [8]. It should be emphasized
that contact load value depends essentially on the number of contacting points.
Some novel dynamical phenomena have been detected. For example, if the upper plate
is ball-type supported, and the lower one is clamped along its contour, synchronization
takes place. Namely, both plates start to vibrate with the same fundamental frequency
ω = 1.6 (a ball frequency) earlier than in the case of clamping and ball-type supports.
After the occurring synchronization, further increase of loading has not changed dynamics qualitatively.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
J. Awrejcewicz and A. V. Krysko, Analysis of complex parametric vibrations of plates and shells
using Bubnov-Galerkin approach, Archive of Appl. Mech. 73 (2003), 495–504.
J. Awrejcewicz and V. A. Krysko, Nonclassical Thermoelastic Problems in Nonlinear Dynamics of
Shells, Scientific Computation, Springer-Verlag, Berlin, 2003.
J. Awrejcewicz, V. A. Krysko, and A. V. Krysko, On the economical solution method for a system
of linear algebraic equations, Math. Probl. Eng. 2004 (2004), no. 4, 377–410.
J. Awrejcewicz, V. A. Krysko, and G. G. Narkaitis, Bifurcations of a thin plate-strip excited
transversally and axially, Nonlinear Dynam. 32 (2003), no. 2, 187–209.
B. Ya. Kantor, Contact Problems of Nonlinear Shells Theory, Naukova Dumka, Kiev, 1990.
M. S. Kornishin, Nonlinear Problems of Plates and Shallow Shells and Methods of Their Solutions,
Nauka, Moscow, 1964.
A. V. Krysko, V. A. Krysko, V. A. Ovsiannikova, and T. B. Babenkova, Complex vibrations of twolayer uncoupled plates subject to longitudinal sinusoidal load, Izv. Vuz., Striotel’stvo (2002),
no. 6, 23–30 (Russian).
A. N. Sharkovskiy, Existence of cycles exhibited by continuous mapping of a straight line into itself,
Ukrainian Math. J. 16 (1964), no. 1, 61–71 (Russian).
J. Awrejcewicz: Department of Automatics and Biomechanics, Faculty of Mechanical Engineering,
Technical University of Lodz, 1/15 Stefanowskiego Street, 90-924 Lodz, Poland
E-mail address: awrejcew@p.lodz.pl
V. A. Krysko: Department of Mathematics, Saratov State University, B. Sadovaya 96a, 410054
Saratov, Russia
E-mail address: tak@san.ru
O. Ovsiannikova: Department of Mathematics, Saratov State University, B. Sadovaya 96a, 410054
Saratov, Russia
E-mail address: tak@san.ru