CC BY
2
Djalilov M.L., Rakhimov R.Kh.
05.14.01 ЭНЕРГЕТИЧЕСКИЕ СИСТЕМЫ И КОМПЛЕКСЫ
COMPLEX POWER SYSTEMS
DOI: 10.33693/2313-223X-2020-7-4-25-30
The compelled fluctuations of a rectangular two-layer piecewise-homogeneous plate
of a constant thickness
M.L. Djalilov1' a ©, R.Kh. Rakhimov2' b ©
1 Fergana branch of the Tashkent University of Information Technologies named after Muhammad Al-Khorazmiy, Fergana, Republic of Uzbekistan
2 Institute of Materials Science, SPA "Physics-Sun", Academy of Science of Uzbekistan, Tashkent, Republic of Uzbekistan
а E-mail: mamatiso2015@yandex.ru b E-mail: rustam-shsul@yandex.com
Abstract. This article discusses forced vibrations of a rectangular two-layer piecewise homogeneous plate of constant thickness, when the material of the upper layer of the plate is elastic and the other satisfies Maxwell's model, that is, viscoelastic. The transverse displacement of points of the contact plane of a two-layer plate is determined, which satisfies the approximate equation obtained in [1], replacing only the viscoelastic operators of the upper layer of the plate with the elastic Lames coefficients, respectively. Fluctuation rectangular is free опертой a piecewise-homogeneous plate at nonzero initial conditions, frequencies of own fluctuations are calculated, and the analytical decision of this problem is under construction. The received theoretical results for the decision of dynamic problems of cross-section fluctuation of piecewise homogeneous two-layer plates of a constant thickness taking into account viscous properties of their material allow to count more precisely cross-section displacement of points of a plane of contact of plates at non-stationary external loadings.
Key words: vibrations, two-layer plate, displacements, elastic, viscoelastic, boundary conditions, initial conditions, operator, Maxwell's model, differential equation, hinged plastic, complex frequency, Poisson's ratios, Fourier series, oscillation equations
FOR CITATION: Djalilov M.L., Rakhimov R.Kh. The compelled fluctuations of a rectangular two-layer piecewise-homogeneous plate of a constant thickness. Computational Nanotechnology. 2020. Vol. 7. No. 4. Pp. 25-30. DOI: 10.33693/2313-223X-2020-7-4-25-30
DOI: 10.33693/2313-223X-2020-7-4-25-30
Вынужденные колебания прямоугольной двухслойной кусочно-однородной пластинки постоянной толщины
М.Л. Джалилов1 ©, Р.Х. Рахимов2, b ©
1 Ферганский филиал Ташкентского университета информационных технологий имени Мухаммада Ал-Хоразмий,
г. Фергана, Республика Узбекистан
2 Институт материаловедения Научно-производственного объединения «Физика-Солнце» Академии наук Республики Узбекистан,
г. Ташкент, Республика Узбекистан
а E-mail: mamatiso2015@yandex.ru b E-mail: rustam-shsul@yandex.com
Аннотация. В данной статье рассмотрена вынужденные колебания прямоугольной двухслойной кусочно-однородной пластинки постоянной толщины, когда материал верхнего слоя пластинки упругий, а другой удовлетворяет модели Максвелла, то есть вязкоупругий. Определено поперечное смещение точек плоскости контакта двухслойной пластинки, удовлетворяющий приближенному уравнению, полученному в работе [1], заменяя только вязкоупругие операторы верхнего слоя пластинки на упругие коэффициенты Ляме соответственно. Для прямоугольной, свободно опертой кусочно-однородной пластинки при ненулевых начальных условиях, вычисляются частоты собственных колебаний, и строится аналитическое решение этой задачи. Полученные теоретические результаты для решения динамических задач поперечного колебания кусочно-однородных двухслойных пластин постоянной толщины, с учетом вязких свойств их материала, позволяют более точно рассчитывать поперечное смещение точек плоскости контакта пластин при нестационарных внешних нагрузках.
Ключевые слова: колебания, двухслойная пластинка, смещения, упругий, вязкоупругий, граничные условия, начальные условия, оператор, модель Максвелла, дифференциальное уравнение, шарнирно опертая пластика, комплексная частота,
коэффициенты Пуассона, ряды Фурье, уравнения колебания
f ^
ССЫЛКА НА СТАТЬЮ: Джалилов М.Л., Рахимов Р.Х. Вынужденные колебания прямоугольной двухслойной кусочно-однородной пластинки постоянной толщины // Computational nanotechnology. 2020. Т. 7. № 4. С. 25-30. DOI: 10.33693/ 2313-223X-2020-7-4-25-30
V J
Let us consider the case of forced vibrations of a rectangular two-layer piecewise homogeneous plate of constant thickness, when the material of the upper layer of the plate is elastic, and the other satisfies Maxwell's model, that is, viscoelastic.
We reduce the problem to finding the transverse displacement of the points of the contact plane of a two-layer plate, which satisfies the approximate equation obtained in [1], replacing only the viscoelastic operators and by the elastic Lame coefficients and, respectively:
—r 1 + Q,| Д—г- 1 + Q, U2W) + Q. I —¡r-
-ч dt4 j+q ^ dt 2
dt6
+о5!Д^|+Об IД2dW-|+Q7 (Д3 W) = ^(x, У, t).
dt4
, d2W
dt
(1)
Where operators Q. are defined under the formulas (2), received in work [1].
If a two-layer plate occupies a region 0 < x < l, 0 < y < l2 and its four edges are hingedly supported, then in this problem the boundary conditions for the displacement W have the form:
„, d2W d4W „ , „ , , W = —- =-- = 0 (x = 0; x = l1 );
dx dx
(2)
W=
dy2 dy4
= 0 (y = 0; y = I2 ).
The compelled fluctuations of a plate are raised by nonzero initial conditions, that is
W = ф(х, у ); ^ = ^(х, у ); dy
d2W d'W
(3)
dy2
dy7
= 0.
Thus functions ^ also ^ should satisfy to type (2) conditions, that is
^ = 0; ^ = 0 (x = 0; x = y = 0; y = l2). Let's result the equation (1) in a kind
M3(g + M?(y + M1(Z3) + Z4]( W ) = 0,
(4)
d W d4W
+
Djalilov M.L., Rakhimov R.Kh.
where the operators Z are equal:
where are - viscoelastic operators, which for the Maxwell model have the form:
Z = A; TT + a
— , A__+ A _•
-, 4 14 -. 2 - 4 18 - 6 '
дх dt dx дх
= д4 д4 д6 д6 d6 Z2=A atW +A 5 +Al0 +Al5 Ä4+Al7 дх^
r = A il A д4 A A iL A д6 (5)
Z3 = Al дt4 +A2 дt2дх2 +A4 дх4 +A7 дt6 +Al1 дt4дх2 + A д6 A д6 A д6 ; +Al3 дХ6+Al3 +Al6
r = A il A д6 д6
Z4 = A8 - 6 + A ■ - + A
4 8 n 6 -.4- 2 12 -.2- 4'
дх д€дх дгдх
and the coefficients A are:
M, (Ç) = ^ к-^je-ï-Z)Ç(0d^ l T о
M (ç)=^2 fç- 2 je-(t
l T о
+4 je-(t-Z)(t-^ZUK T 0
[ 31
M-(0 = ц- Z--je-(t l T о
+A je-(t-Z)T(t-^(^--L je-(t-Z)(t
T i 2T j„
(7)
A =(лоРо +hiPi )2;
Substituting (7) in (4), after some transformations we will A2 =-2ц0(/)0D0(h0р0 + h2 p1 ) + h2 D1 (h0D0 + h2 D1 ) + h0p0(h0 + h2 )); receive the equation of fluctuation of a piecewise-homogeneous
two-layer plate in the differential form
A = -2(hi D1 (h0 P0 + hiPi ) + h0 P0 (h0 D0 + hiDi)- h0 P0 (h0 + hi )) ; A =Ц0 ( D0 ( + 2hi));
A = -4^0(D0 ( + hi) + hi(hD0 -hiDi));
A =-4^ Di;
A=7 Ho1hi2 P0 (332 P2 + h0 P0 (h0 P0+4hi Pi )) (2 - D0 ) ;
A = 7 h Pi ( p0 + hi Pi (hi Pi + 4h0 P0 ))(2 - Di );
A = -7 Pi (hi2 PiDi (2 - Di ) + 6h0 P0 (4D0 + (i - Di )(2 + D0 )) + + 4h0 hi (P0 (2 - Di ) + PPD (4 - Di ))) ;
Ai0 = -7 И-2 h0 P0 (-h0 P0 (4 + d0 ) + 6h0 Pi (3 - D0) + + 2h0 hi P0 (2Di (4 + Di )-(4 - 3D0 ))) ;
Ai = -7(h04 P0 (8D0 -9DH + 6) -h?P2 (7Di2 -6Di -2) +
+ 6h0 h (P0 Pi (3Di-2(i + D0 ) ) -+ PÔÔ + P2); (6)
A2 = ! И0 h0 hi2 Pi D0 (9h0 ( - Di ) + 4hi (2 - Di )) ;
A3 = 3 И0 (h0 P0 (( - 3D0 + 2) + h4 Pi Di (3 - Di ) + + 3h0hi2(P0((3 + 2D) + 2(-Di) + P(6D0Di-6D0 -D))--2h0hi (p0 (2h0 ( -2D0 -i) -h2 (2 + Di) --4pi D0 (h0 + h°2 (3Di -i)))); A4 =2Ц^2 h0 h2 P0 Di (i-D0 ); A5 = 3 h0 P0 (5D0 + 11D0 + 2) - h4 pDi (l - 7Di ) + + 3h0 h2 (P0 ( (l + D )-(2 - D0 )) D (5-2D0 )) + +2h0 hi P0 (h (( - 2D0 - i) - 2Di (1 + 2D0 )) - hi2 (2 - 3D ));
Ai6 =-2 ^ h0D0 (5h0 - 3h0 h2 (2D ( + D )) + 4hi (h0 - h (2 + Di )));
A7 = \ И0 (h04 D0 (4D0 + 5) + h4 D + 3h0 h2 (( Di - 3D0 - D ) + -4h0 hi D0 (h0 (2D-i)-hi2 (2-Di )));
As =2 hi2 ( D (l + 4Di)- 3h2 ),
54W
11 dt4 J 2 Г dt2
+ G2 + G3 (A2W ) + G4 [dW >
+ J + G6lA2l + G7
+ ga
( 2 d2W
dt
2 | + Gio (A3W ) + Gi
id6 W dt6
d7W
+ G 8
' д4 W ^ A—r- | +
dt4 J
dt '
T | + G12
Ad!WW 1+ (8)
dt5 J
+ G,:
[ , 2 d3W ^ _ ^ 3 dW . _ A"dtT J + G14 "dW | + G15
4
[d8 W dt8
+ G.
f d6 W
dt
6 | + G17
^л2 d4W . „ A2-- 1 + G..
\ " у \ where the coefficients G. are:
dt4
( 3 d2W
dt2
= 0,
111
Gi Al; g2 = A; G3 = a4;
T T T
1
1
1
g4 =Ai A; g5 = A3a2a9;
I 11
Ge =^2" A5 + A4 +—A12;
T TT3
II
G7 = — A7 + A1 + ^ T A8;
TT
1 1
G8 =^1 A1 +^2A3 + ^T A9 A1i;
TT
1
1
G9 = ^1A4 + Hi A5 + Hi A6 + 3~ A12 + Hi — A13;
TT
= 1 . = 2 1 .
G10 = M-1-2 A16; G11 = 2 M-1-A7 + A8; T TT
1 2 1 1
Gii — i A9 + 2 Hi A10 + 2 Hi Ац ;
T T T
111
G13 — i—A12 + 2^-A13 + 2h2~ A15;
TTT
1 1
G14 = 2 A16 + H2_ A17; G15 = HiA7 + A8; TT
G16 — A9 + HiAi0 + HiAii;
G17 — A12 + H1A1i + H1A14 + HiA15;
G18 — H1A16 + Hl A17 + H1A18.
(9)
6
6
4
+
DEVELOPMENT OF NEW ENERGY UNITS BASED ON RENEWABLE KINDS OF ENERGY
Owing to edge conditions (2) required displacement W and we will search in a kind
W = X X Wn, m(t)sin(Ynx)sin(vmy),
(10)
Where n, m - numbers of harmonics of fluctuation in corresponding directions x, y. Solution (7) will satisfy boundary conditions (2) if
_ nn _ nm
Yn = j ' Vm = j ' l1 2
where l1 and l2 - the geometrical sizes of a plate.
Substituting expression (10) in the equation (8), for Wn m(t) it is received the ordinary differential equation
(11)
where coefficients at derivatives are equal:
B1 = ^i; B2 =±- f G7-XG16
Gn I hn
B:
3=11g4 - h2 G12
b=G5 [G - h2 G1s - h2 Gi7
1 y
B5 = 1 Y Gi5 h2 G5 - hl G13 ) ;
1 -G15 f Y G h 1G-- Y G 2 Y2 + 1T G п. ho4 18
G6 - hiG» ) ; Bs = 1 Y2 ( G15 ho4 l
(12)
and
' nhn )2 ( nm )2
T-'f " ) ПТ );
for the decision of the equation (11) we search in a kind
'bn
Wn, m = W0exp\ h- &
(13)
where ; - dimensionless, generally, the complex frequency, which valid part characterizes attenuation of fluctuations of a plasticity, and an imaginary part of own fluctuations of a plate.
Substituting expression (13) in the equation (11), for complex frequency ; it is received the algebraic equation of the eighth order
+ B^7+B'2z,6 + B'^5+b;^;+B^3+B^2+B'= 0, (14)
where
b' = ^; b2 =—\g;-xg'16
G'
B4=GT
g15
G'
1
B3=äG4- i G'12
G'i-TT G'i8 — TT G'l7
b=- -l x 5 g'5 h2a
G5 — TT G1s
b = 1 Y
G15 h0
G2 - h~2 G9 + TT G18
V h0 h0
B7=i h4 lG6 g'T
b = 1 y2'
G15 h0
G3 - T2 G10
V h0
(16)
G'= % A' ; G2= P22A2 ; G3= P2X ; g; = 2p22c a'+p22ca8 + p23ca8 ;
G'5 = C _1A' + P2 C _1A' + P22CA9 ; G6 = P2 (2P22C-1a; + C+ P^CA'12 ); G7 = P22 (c ~2 a; + A' + 3P2 A' ); G8 = P22 (c ~2 A2 + C-2A3 + 3P22A9 + P2 AÎ1 ); g9 = p22c ~2a;+p2c ~2 a; + c ~2 a' + 3p23a'2+p22a'3; G'o = P22A1e; G'1 = C-1 (2A' + 3P2 A'); G'2 = P22C ^ (3P22 A'+ A'o + 2P2 A'1 ); G'3 = P22C _1 (A + 2P2A'3 + a;5 ); G'; = p22C-1 (A + a;7 ); G'5 = P22C -2 (A' + P2A8 ); G'e = P22C (P22A9+ A'o + PX );
G17 = C (P2A12 + P2 A13 + + P2A15 ); G18 = C (P2 A'6 + P2A^7 + A^8 ) '
where coefficients Aare equal:
^'=(1 + ^p)2; A'' =-2 (D0 (1 + hp) + hp(D0 + hD1 ) + (l + h )); A3= 2 (hD1 (1 + hp) + (D0 + hD1 )-(l + h)); A4 = 4Do (1 + hp); A5 = -4 ( (1 + h) + h (Do - hDi
A6 =-4h2Di; A' =1 (3h2p2 +(1 + 4hp)(2 - Do ));
A8=1 h2p(3 + hp(hp + 4 ))(2 - D1 );
A9=-^h2p(h2pD1 (2 - D1 ) + 6(2 (1 + Do ) + D1 (2 - Do )) + +4h (2 - D1 ) + pDo (4 - D1 )) ;
A'o =-1 (-(4 - D2 ) + 6h2p(D1 (3 - D1 ) + + 2h (2D1 (4 - Do )-(4 - 3Do ;
A'1 = -7(8Do -9Do2 + 6)-h4p2 (7D2 -6D1 -2) + + 6h2 (p(3D1 - 2 (1 - Do )) +1 + p2 ) +
+2h ((4 - 3Do ) + 2p ((2 + 4Do - $ ) + h2D1 (5 - D1 ; A'2 = 3 h2p2Do (9 (1 - D1 ) + 4h(2 - D1
A'3 = 3((15Do - 3D2 + 2)-h4p2D1 (3- D1 ) + + 3h2 ((/^^ (3 - 2D1 ) + 2 (2 - D1 )) + p (16DoD1 - 6Do - D1 )) ^
- 2h ((^ (D2 - 2Do -1) - h2 (2 - D1 )) - 4pDo (1 - h2 (3D1 -1)));
(17)
n = 0 m = 0
B
6
7
G15 h0
Djalilov M.L., Rakhimov R.Kh.
A'4 =-2hD (1 - D0 ); a;5 = I (-(1ID0 + 5Do2 + 2)- h4pDi (1 - 7Di ) + + 3h2 (( (1 + Di ) - (2 - Di )) + pDi (5 - 2Do )) +
+ 2h (2 (Do2 - 2Do -1) - 2Di (i + 2Do ) - h2 (2 - 3Di )));
A'e = -3 Do (5 - 3h2 (2Di - 3) + 4h (i - h2 (2 - Di ))) ; (17)
A'y = 2(Do (4Do + 5)-h4Di - 3h2 (ioDo Di - 3Do - Di) + + 2hDo ((2Di -1)-h2 (2 - Di ))) ;
A'8 = 3 h2Di (h2 (1 + 4Di )-3), also dimensionless parameters are entered
h = ïhL; p=Pl; p2 c = _Üb
P 0
b = f; Do b 0
2(-v 0 )
H 0 D,
t b 0
2 (1 -v ! )
Here v0, v1 - coefficients of Poisson of a material of layers of a plate; b0, b1 - speeds of distribution of cross-section waves of a material of layers of a plate; c - parameter considering time of a relaxation t, thickness of the top layer h0 of a plate and speed b0.
Proceeding from Hurwitz theorem [4] and positivity of factors (15), follows that the valid parts of complex roots of the algebraic equation (14) are negative that responds fading character of fluctuation of a two-layer plate, and imaginary parts characterize frequencies own fluctuations of a two-layer plate.
Let's designate roots of the algebraic equation (13) in a kind
£1 = Re£i0; = Re'^o; ^3, 4 = ReU ±iIm^20;
£5, 6 = Re £30 ± ' Im£30; 8 = Re £40 ± i Im £40.
Thus Re j attenuation coefficients, and Im j - frequencies of own fluctuations.
Proceeding from character of roots of the characteristic equation, the problem common decision we can write down in a kind
W„: m = e li{) tC1 + e 1(2 tC2 + +eJ>t [C3 cos ( J) + C4 sm(a„, J )] + +e-s("' "kt [C5 cos (|3n, mt) + C6 sin (|3n, mt)] + +eJt [Cy cos (y „, J) + C8 sin(y „, J)], where designations are entered:
(18)
X(n, m) . °1(1) ■
-^0-1^ f ■ Я(", m) ■ Re f10' °1(2) ■
"T^Re'fn
S(",m) = -—Re f : (j = 2, 3, 4);
1 h ^jo
0
an, m = br1m f2o; ßn, m = T"1™ ^ Yn, m = Ь0|тf40.
h0 h0 h0
(19)
For definition of constants C. of function ^(x, y) also ^(x, y) we will spread out to double seriates of Fourier
ф(л y )=Z Z^n. m Чуx I sinl"T y I ;
n = 1 m = 1 V l1 J V l2 J
(20)
\ x-1 X-1 • • 'bill
У )=Z Z^n .m sln!-p X I Sln!"P У |.
n = 1 m = 1 V l1 J V l2 J
C - 2S(n' m)ß C
5 zu3 Pn, mL 6
Satisfying the decision (7) with expressions (18) both (19), and initial conditions (3), for C we will receive system of the algebraic linear equations:
Ci + C2 + C3 + C5 + C7 =9„, m ;
-S(n, m)C -S(n, m)C -S(n, m)C +a C s(n, m)C + 01(1) C1 01(2) C2 02 C3 +Kn, mC4 03 C5 +
+ Pn, mC6 04 , C7 +Yn, mC8 =¥n, m;
( )m))2 C-(s^)2 C2+[(s^ m))
- m)a„, mC4 + [(s("' m))2 + P" ' -[(Si"' m))2-Ym]C7 -2S"m)Yn,mC8 = 0;
(8(^;)m))3Cl -(S(^2)m))3C2 +S("'m)[(8("' m))2 +a2, f -an,m[3(m))2 -a2„, m]C4 + 8(n-m)[(m))2 + pn, m]C5 -- Pn, m D^11, m))2 -p2n, m ]C6 + 84", m) [(8l", m))2 - 3Y2n, m ]C? --Yn, m [3(8l1, m))2-Y2", m ] C8 = 0; (m))4 c1 -(8(î,">)4 c2 +[(5(", -6-))2a2„, m +a4, m]c3 --S("' m)a„, m [(s^' m))2-a2„, m ] c4 + + [(8(. m))4 -6(s(n m))2 P2„' m +P4' m]c5 -
-s("' m)P„' m [(s("' m))2-Pi m ] c6 +
+ [(s(."' m))4 - 6(si"' m))2 Yn, m +y2' m ]c7 -
-s4n' m)Y"' m [3(s4"' m))2-Y2"' m ] c8 = 0; (s&m))5 Cx-(8(J2 r')5 ^2 +
+ 8(". r)[(g(". r))4 - 7 (g(n. r))2 a2
-a„, m[2(82"'m))4 -7(m))2al m + a4, m]C4 +
+ 8(n, r)[(s("' r))4 -7(t. r))2p2' r + 2p4' m]C5 -
-Pn, m [2 (si"' r))4 - 7(8("' r))2 p2, m+P4, m ] C6 -
- 8("' r) [(84"' r) )4 - 7(84"' r))2 y2, m + 2Y4, m ] C7 -
- Y", m [2 (8^ m))4 - 7(84" r))2 Y", m - Y4, r ] C8 = 0;
(S&m))6 Cx-(Si?2)m))6 C2 + +[(("'m))6 -9(S("' m))4 a2, m + 9(s(,n'm)
- S(,n' m)a„. m [3 ( m) )4 -14 (( m) )2 a2, m + 3a4, , + [("' m))6 - 9 (s("' m))4 P2, m + 9 ( m))2 P4, m -P6, m JC5 -
- m) P„, m [3 (( m) )4 -14 ("' m) )2 P2, m + 3P4, m ] C6 -
-[(84"' m))6 - 9 (8(2, m))4 y2, m + 9 (842' m))2 Y 4, m -y6, m J C7 -
(21)
^ -
m ^n, m
1G +
5(n, m)
fn, m [3 (84" m))4 -14 (П m))2 Y2n, m + 3YП, m ] C8
= 0;
0
0
0
DEVELOPMENT OF NEW ENERGY UNITS BASED ON RENEWABLE KINDS OF ENERGY
(sS,m))7 c-( ;>)7 c2 +
Cc -
+s2"'m)[(("' m))) -12m))4 al m + 21 (m))2 a:, m -4a6, m] -a„, m [:(s(", m))) - 2l(s(, m))4 a2„, m +12 ( m))2 a4, m-a^, m ]« -S3", т)[(3", m))) -12 m))4 ß2n, m + 2l((", m))2 ß4, m - 4ß), m ] -ß", m [4(S(", m))6 -2l(s(", m))4 ß2„, m + 12(S(", m))2 ß4, m-ß) m ]С) -- s4", m)[(s(4", m))) -12(( m))4 Y2„, m + 21(S(", m))2 y4, m - 4y), m ] С7 -", m [4(S(", m))) -21(( m))4 y2, m + 12(S(", m))2 y4, m-y4, m ]Q = 0.
-Y
The general solution of problem (18) together with (21) gives a solution to the problem of forced vibrations of a rectangular hingedly supported piecewise-homogeneous two-layer plate.
FINDINGS
1. When solving the problem of vibrations of a rectangular hingedly supported two-layer plate, when the material of the lower layer is viscoelastic (Maxwell's model), the obtained frequency equation has complex roots, the real part of which is negative and characterizes the damping of the plate vibration, and the imaginary part determines the natural frequencies of the vibration.
2. An analytical solution to the problem of vibrations of a rectangular hingedly supported two-layer plate is obtained, which makes it possible to evaluate the influence of viscosity and two-layer structure on the parameters of the plate vibration.
References
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Статья проверена программой Антиплагиат
Рецензент: Раджапов С.А., доктор физко-матматических наук; ведущий научный сотрудник Физико-технического института НПО «Физика-Солнце» Академии наук Республики Узбекистан
Статья поступила в редакцию 15.11.2020, принята к публикации 20.12.2020 The article was received on 15.11.2020, accepted for publication 20.12.2020
ABOUT THE AUTHORS
Mamatisa L. Djalilov, Cand. Sci. (Eng.); Head at the Department "Computer Systems" of the Fergana branch of the Tashkent University of Information Technologies named after Muhammad Al-Khorazmiy. ORCID: https:// orcid.org/0000-0002-9471-4893; E-mail: mamatiso2015@ yandex.ru
Rustam Kh. Rakhimov, Dr. Sci. (Eng.); Head at the Laboratory No. 1 of the Institute of Materials Science "Physics-Sun" of the Republic of Uzbekistan. Tashkent, Republic of Uzbekistan. ORCID: https://orcid.org/0000-0001-6964-9260; E-mail: rustam-shsul@yandex.com
СВЕДЕНИЯ ОБ АВТОРАХ
Джалилов Маматиса Латибджанович, кандидат технических наук; заведующий кафедрой «Компьютерные системы» Ферганского филиала Ташкентского университета информационных технологий имени Мухаммада Ал-Хоразмий. Ташкент, Республика Узбекистан. ORCID: https://orcid.org/0000-0002-9471-4893; E-mail: mamatiso2015@yandex.ru Рахимов Рустам Хакимович, доктор технических наук, профессор; заведующий лабораторией № 1 Института материаловедения Научно-производственного объединения «Физика-Солнце» Академии наук Республики Узбекистан. Ташкент, Республика Узбекистан. ORCID: https://orcid.org/0000-0001-6964-9260. E-mail: rustam-shsul@yandex.com
