ANALYSIS OF THE GENERAL EQUATIONS OF THE TRANSVERSE VIBRATION OF A PIECEWISE UNIFORM VISCOELASTIC PLATE Текст научной статьи по специальности «Физика»

05.14.01 ЭНЕРГЕТИЧЕСКИЕ СИСТЕМЫ И КОМПЛЕКСЫ

COMPLEX POWER SYSTEMS

05.13.06 АВТОМАТИЗАЦИЯ И УПРАВЛЕНИЕ

ТЕХНОЛОГИЧЕСКИМИ ПРОЦЕССАМИ И ПРОИЗВОДСТВАМИ

(ПО ОТРАСЛЯМ) (ТЕХНИЧЕСКИЕ НАУКИ)

AUTOMATION OF MANUFACTURING AND TECHNOLOGICAL PROCESSES

DOI: 10.33693/2313-223X-2020-7-3-52-56

Analysis of the general equations of the transverse vibration of a piecewise uniform viscoelastic plate

M.L. Jalilov1' 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

a E-mail: mamatiso2015@yandex.ru b E-mail: rustam-shsul@yandex.com

Abstract. This article discusses the analysis of the general equations of the transverse vibration of a piecewise homogeneous viscoelastic plate obtained in the "Oscillation of inlayer plates of constant thickness" [1]. In the present work on the basis of a mathematical method, the approached theory of fluctuation of the two-layer plates, based on plate consideration as three dimensional body, on exact statement of a three dimensional mathematical regional problem of fluctuation is stood at the external efforts causing cross-section fluctuations. The general equations of fluctuations of piecewise homogeneous viscoelastic plates of the constant thickness, described in work [1], are difficult on structure and contain derivatives of any order on coordinates x, y and time t and consequently are not suitable for the decision of applied problems and carrying out of engineering calculations. For the decision of applied problems instead of the general equations it is expedient to use confidants who include this or that final order on derivatives. The classical equations of cross-section fluctuation of a plate contain derivatives not above 4th order, and for piecewise homogeneous or two-layer plates the elementary approached equation of fluctuation is the equation of the sixth order. On the basis of the analytical decision of a problem the general and approached decisions of a problem are under construction, are deduced the equation of fluctuation of piecewise homogeneous two-layer plates taking into account rigid contact on border between layers, and also taking into account mechanical and rheological properties of a material of a plate. 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 the is intense-deformed status of plates at non-stationary external loadings.

Key words: analysis, approximate, vibrations, two-layer plate, boundary value problem, stresses, deformation, oscillation equations

f \ FOR CITATION: Jalilov M.L., Rakhimov R.Kh. Analysis of the general equations of the transverse vibration of a piecewise uniform viscoelastic plate. Computational nanotechnology. 2020. Vol. 7. No. 3. Pp. 52-56. DOI: 10.33693/2313-223X-2020-7-3-52-56

V J

Jalilov M.L., Rakhimov R.Kh.

DOI: 10.33693/2313-223X-2020-7-3-52-56

Анализ общего уравнения поперечного колебания кусочно-однородной вязко-упругой пластины

М.Л. Джалилов1, а ©, 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

a E-mail: mamatiso2015@yandex.ru b E-mail: rustam-shsul@yandex.com

Аннотация. В данной статье рассмотрен анализ общего уравнения поперечного колебания кусочно-однородной вязко-упругой пластинки [1]. В настоящей работе на основе математического метода, строится приближенная теория колебания кусочно-однородных пластин, основанная на рассмотрении пластинки как трехмерного тела, на точной постановке трехмерной математической краевой задачи колебания при внешних усилиях, вызывающих поперечные колебания. Общие уравнения колебаний кусочно-однородных вязкоупругих пластин постоянной толщины, описанные в работе [1], сложны по структуре и содержат производные любого порядка по координатам x, y и времени t, и поэтому не пригодны для решения прикладных задач и проведения инженерных расчетов. Для решения прикладных задач вместо общих уравнений целесообразно пользоваться приближенными, которые включают тот или иной конечный порядок по производным. Классические уравнения поперечного колебания пластинки содержат производные не выше 4-го порядка, а для кусочно-однородных или двухслойных пластин простейшее приближенное уравнение колебания является уравнением шестого порядка. На основе аналитического решения задачи строятся общее и приближенное решения задачи, выводятся уравнение колебания кусочно-однородных двухслойных пластин с учетом жесткого контакта на границе между слоями, а также с учетом механических и реологических свойств материала пластинки. 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 напряженно-деформированное состояние пластин при нестационарных внешних нагрузках.

Ключевые слова: анализ, приближенный, колебания, двухслойная пластинка, краевая задача, напряжения, деформация, уравнения колебания

ССЫЛКА НА СТАТЬЮ: Джалилов М.Л., Рахимов Р.Х. Анализ общего уравнения поперечного колебания кусочно-однородной вязко-упругой пластины // Computational nanotechnology. 2020. Т. 7. № 3. С. 52-56. DOI: 10.33693/2313-223X-2020-7-3-52-56

The general equations of oscillation of piecewise homogeneous viscoelastic plates of constant thickness, described in [1], are complex in structure and contain derivatives of any order with respect to x, y coordinates and time t, and, therefore, are not suitable for solving applied problems and performing engineering calculations.

To solve applied problems, instead of general equations, it is advisable to use approximate ones that include one or another finite order in derivatives.

The classical equations of transverse vibration of a plate contain derivatives of no higher than 4th order, and for piecewise homogeneous or two-layer plates, the simplest approximate equation of vibration is a sixth order equation.

If in the operators (1.3.8) given in [1] we restrict ourselves to the first two terms, then from equation (1.3.11)

L1{W2) = F^x, y, t),

where are the operators L1 and F1(x, y, t) equal to:

L1 = (M1 (n)K2 (n) - M2(n)Kl(n))(H3(n)E4 (n) - H4 (n)E3( n)) + + (Ml( n)K3( n) - M3( n)Kl(n))(H4 (n)E2(n) - H2( n)E4 (n)) + + (M1 (n)K4(n) -M4(n)Kl(n))(H2(n)E3(n) -H3(n)E2(n)) -(M2(n)K3(n) - M3(n)K2(n))(H4(n)El(n) - H1(n)E4(n))--(M2(n)K4(n) -M4(n)K2(n))(Hl(n)E3(n) -H3(n)El(n)) + + (M3(n)K4(n) -M4(n)K3(n))(Hl(n)E2(n) -H2(n)El(n));

Т. 7. № 3. 2020

Computational nanotechnology

53

F1 - "[Kl(n) (H2(n)E3(n) - H3(n)E2(n) ) + K2(n) ((-)E- (-) - H C-E

3(n)El(n) Hl(n)E3(n) ) +

+K3(n) (Hl(n)E2(n) - H2(n)El(n) )] (( (f)} + + [Ml(n) (H2(n)E3(n) - H3(n)E2(n) ) + M2(n) (H3(n)El(n) - Hl(n)E3(n) ) +

+M3(n) (Kl(n)E2(n) - K2(n)El(n) )] (( ((2(1) )} + " (Ml(n) (K2(n)H3(n) " K3(n)H2(n) ) + M2(n) (K2(n)Hl(n) " Kl(n)H2(n) ) +

fi+fT

3x dy

+M3(n)(Kl(n)H2 (n)" K2(n)Hl(n)]D

we obtain the approximate integral-differential equation

Qi I ~~W 1 + qJa^W I + Q3 (A2W) + Q№

' dt4 d4W

dW

dt2

dW

dt6

+Q5 [A^J + Q6 [A2dWW | + Q7 (A W) = Fl (x, у, t).

(1)

where are the operators Q. and F1(x, y, t) equal to:

Qi = M-2 ( po + hi Pi )2; Q2 = -2M-2 (2 (ho P2 Do + hi Di)(ho Po + hi Pi) + + (P2 - i) (ho Po (ho + hi ) - (ho2 Do Po + hi Di Pi ))); Q3 = 4 ( P2 - i) (ho2 P2 Do + hi Di + hi Di + 2ho hi P2 Do ) ;

Q4 = -i Mi-2 (h02 Po Moi ( P2 + ho Po (ho Po + 4^ Pi )) x

x (2 - Do ) + h° Pi M-1 (3ho2 po + hi Pi (hi Pi + 4ho Po )) (2 - Di )); Q5 = - £M-2 (ho2 P2 po Mo2 (2P2 (4Do (i - Do ) + (P2 - i)(4 + Do2 )) ■

-hi4 P2 M-2 (2 ( - 4Di - i) - (P2 - i) Di (2 - Di )) +

+6ho2 h2 (po Pi Moi M-1 (4 (( Do + Di ) + (P2 -1) x x(2P2 (i - Do )-P2 Di (2 - Do ) + Di (i + Do ))) + M-1 (po +pi )) + +2P2 ho hi (2po Pi Mo iM1-i ) (ho2 (2 + 4Do - d2 ) + +h (2P2 - P2 D1 + 5D1 - D°2 )) + ho2 h Mo 2 x x ((P2 -1)(4 - 3Do ) + 2Di (4 - Do )) + 2h2 p2 M-2 Do (4 - Di ));

(2)

F (x, y, i) = M-2 JL((^0po + hp1 )((-/f)) + h)x

' ......... /£> 32/Л^ (3)

X

h1 Pi

V v

д /-0)+£2/ц i+ho Po

dx2 dy2

dx2

+ (h0 Do Po + hi2 Di ph

d2 e+д2 /yZ

dx2 dy2

dy2 J

dx2 dy2

-2Д

(2M-2 (( P2 Do + hi Di ) ( - Mi /2(1) )) + 2P2 ho hi ) x

DM-1

^ f(o) a2 /(oM

d /xz | + dim-1

3x2 3y2 J

32, d/

3x2 3y2

+Mi-i (ho2 P, Do + hi Di )

,2*(o) я2 /Г02/^ a2/y(l)Vl

1+

3x2 3y2 J

3x2 3y2

If the plate is homogeneous, and W - is the transverse displacement of the points of the "middle" surface - the plane of the plate, then in this case the dependences are satisfied

N = M = Mi; P2 = 1; ho = hi; Cc = ci; Dc = Di. and equation (1) goes into equation

((1 - Co )2 ^+(1 + Co )2 a))+A) +

+hi2 (( (X21+A)2 )+D 0)+4X10)(X20)+A)^!(^)=

M

2

0 3i2

(f ) + ho

f) , 32 fy

(0) лл

3x2 3y2

(4)

(f )+h

d2 f (1) a2 AlAW

v Jxz +_Jy_

'0 dx2 dy2

where on the left is the product of two operators: the first describes the process of longitudinal oscillation, and the second describes the transverse vibration.

The approximate equation from the general equation (1.3.12) given in [1] is introduced similarly and we obtain for dU1 dV\ dy dx

34

+ G2A + G3 — + + G5A2 + G6^~ +

1 dt2 2 3 dt44 dt2 5 6 dt6

i! dt2

(5)

(6)

+G7A + GsA2 + GgA3 -Ц) = F ( y, t ),

where are the operators Q. and F1(x, y, t) equal to:

Gi = M-1 (h0 po + hi Pi ); g2 = (ho +hi ); G3 =1 M-2 (ho2 (ho Po + 3hi Pi ) Po Mo1 + h (hi pi + 3ho Po ) Pi M-1 );

G4 = (ho2 ( ho Po Moi + 3hi (PO Moi + Pi M-1 )) + +h0 (( Pi Mi1 + 3P2 ho (Po Mo1 + Pi M-1 )));

G5 =1M-2 (ho2 ( ho + 3hi ) + h2 (hi + 3P2 ho

G6 = i0o (ho5 ^2 Po Mo2 (( M-1 + Po Mo1 ) + +h5 p1 M-1 (ioPo Mo 1 + p1 M-1 ) + 5ho h1 Po p1 Mo 1 M-1 x ^ (ho3 Po Mo1 (3 - 3Do - D°° ) - hi3 P2 Pi M-1 (3 - 3Di - D?

G7 = (-13(ho5 ^2 po Mo2 + hi5 P° Mi-2 ) +

+2o (ho5 P2 + hi5 ) po Pi Mo1M- - 5ho hi x x (ho3 Po Mo1 ((3 - 3Do - D2 ) Po Mo1 - (Do - 4) Pi M-1 ) + +hi3 P2 pi Mi-1 ( - 3D^ D2 ) Pi Mi-1 - (Do - 4) pi Mi-1 po Mo1 )));

1

h

0

+

-4Do Mo1 Д

X

+

Jalilov M.L., Rakhimov R.Kh.

G = l2o (23( p2 p2 mq1 + P2 M-2 )+

+10 ( P2 p1 M-i + p0 Mo 1) + 5ho h1 x x(ho3 (pi M-1 - (Do -4)po Mo1) + hi4 (po Mo1 - (Di -4)p-M-1

G = i1o (-24 (ho5 P2 po Mo2 + hi5 pi Mi-2) +

+6 (ho5 P2 + hi5 )po p- Mo1 Mi-1 --6hohi (ho3poMo- (l - 3Do - D2 )m0-1 - (Do - 2)piMi-1) +

+hi3P2piMi-i ((3 - Di - Di2 )piMi-i -(Do - 2)piMi-ipoMo1)));

(6)

1 (. У. t ) = P2

(0)

dx2

dy

+ N-

/ d2 /У

(1) ll

dx2

dy2

/

1

+

2

P2 hi p1 M1

ч

l52/Х0) d2/<0)l

-h02 po M0-1

N1-

, 0 dx2

1 d2/(1) d2 /У

dx2

yz

dy2 .J

(1) ll

yz

dy2

dt2

l/ d2/y

02 dx

2 (0) yz

dy2

. d232/ЛЪ2

dx2

dy2

dx2

Despite the fact that equation (1) is approximate, it is quite complicated. The operators (2) contain all parameters and operators characterizing both the mechanical and rheological properties of the piecewise homogeneous plate material and its geometric dimensions.

Approximate equation (1) is simplified in particular cases when solving specific oscillation problems. For example, operators (2) are greatly simplified when the Poisson ratios of both components are constant, or when the thicknesses of both components are equal, and so on.

For example, if h0 = h1 and v0 = v1, then the operators Q. in (6) have the form:

Qi = M-2 h02 (p0 +pi )2; 2 = -2M-2 ho2 (0 (2 +1) (Po + Pi) + ( +1) (Po - Do (Po - Pi)));

Q3 = 4( -i)ho2Do (ЗР2 +1);

Q4 = -6M-2 ho4 (2 - Do )(po Mo1 (3P2 + Po (Po + 4Pi)) +

+Pi M-1 (3PO +P1 (P1 + 4Po))); (7)

Q5 = - 6 ho4 (P2 Po Mo-2 (4Do (4 - Do) + P2 (8Do (1 - Do) + 5) + + (P2 -1) (12 - 6Do + d2 )) + 2Po P1 Mo1M-1 (2 (6Do + P22 (2 + 5Do) +

+P2 (2 + 9Do - Do2)) + ( - 1)P2 (2 - 3Do + Do2) + Do (1 + Do)) + +P° M-2 (8 (1 + Do - Do2) + 4P2 Do (4 - Do) + ( -1) Do (2 - Do)));

Q = 3 ho2 (Po Mo-1 (4P2 Do (2 + 5P2 - 3Do (P2 -1)) + + (P2 -1) )P2 (2o - 8Do - 13Do2) + 6Do (1 - Do))) + + P1M-1 Do (4 (4 + Do) + 4P2 (4 + 2P2 + 5Do) +

+17(P2 - 1)(Do + 2P2 (1 - Do

The sixth order operator in equation (1) can also be represented as the product of second and fourth order operators if the plate is elastic and the coefficients Q. connected by addiction

QQQ = QQQ + QQQ.

For a two-layer elastic plate with given parameters of its components, relation (7) gives a 10th order algebraic equation with respect to the relation hjhv the sixth-order operator in (1) can be represented as the product of two lower-order operators

d2 d2 )f d2 d2 d4 d4 ,, ч

if the coefficients Q. and A. linked by dependencies

Q1 = A1A2; Q2 = A1A4 + A2A3; Q3 = A2A4; Q4 = A1A5; Q5 = A2A5; Q6 = A1A6; Q7 = A2A6.

FINDINGS

1. The study of vibrations of piecewise-homogeneous plates in an accurate three-dimensional formulation allows us to derive the general and approximate equations of vibration of such plates based on them without using any hypotheses.

2. It is shown that the simplest approximate equation of vibration of a two-layer plate is a sixth-order equation with respect to derivatives describing its longitudinal-transverse vibration.

3. For an elastic two-layer plate, the sixth-order operator splits into the product of the second-longitudinal and fourth-order transverse-wave operators if the thicknesses of the plate components satisfy the derived equation containing the parameters of these components.

4. Formulas are obtained for determining displacements and stresses through the sought-for functions at any point of a two-layer plate.

References

1. Rakhimov R.Kh., Umaraliev N., Djalilov M.L. Oscillations of bilayer plates of constant thickness. Computational Nanotechnology. 2018. No. 2. ISSN 2313-223X.

2. Love A. Mathematical theory of elasticity. Moscow-Leningrad: ONTI, 1935. 630 p.

3. Filippov I.G., Egorychev O.A. Wave processes in linear viscoelastic media. Moscow: Mechanical Engineering, 1983. 272 p.

4. Achenbach J.D. An asymptotic method to analyze the vibrations of elastic layer. Trans. ASME. 1969. Vol. E 34. No. 1. Pp. 37-46.

5. Brunelle E.J. The elastics and dynamics of a transversely isotropic Timoshenko beam. J. Compos. Mater. 1970. Vol. 4. Pp. 404-416.

6. Brunelle E.J. Buckling of transversely isotropic Mindlen plates. AIAA. 1971. Vol. 9. No. 6. Pp. 1018-1022.

7. Callahan W.R. On the flexural vibrations of circular and elliptical plates. Quart. Appl. Math. 1956. Vol. 13. No. 4. Pp. 371-380.

8. Dong S. Analysis of laminated shells of revolution. J. Esg. Mech. Div. Proc. Amer. Sac. Civil Engrs. 1966. Vol. 92. No. 6.

9. Dong S., Pister R.S., Taylor R.L. On the theory of laminated anisotropic shells and plates. J. of the Aerosp. Sci. 1962. Vol. 29. No. 8.

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МНОГОМАСШТАБНОЕ МОДЕЛИРОВАНИЕ ДЛЯ УПРАВЛЕНИЯ И ОБРАБОТКИ ИНФОРМАЦИИ

MULTISCALE MODELING FOR INFORMATION CONTROL AND PROCESSING

Статья проверена программой Антиплагиат

Рецензент: Раджапов С.А., доктор физ.-мат. наук; ведущий научный сотрудник Физико-технического института НПО «Физика-Солнце» Академии наук Республики Узбекистан

Статья поступила в редакцию 15.07.2020, принята к публикации 20.08.2020 The article was received on 15.07.2020, accepted for publication 20.08.2020

ABOUT THE AUTHORS

Mmatmatisa L. Jalilov, 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. Fergana, Republic of Uzbekistan. 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 Ммаматиса Латибджанович, кандидат технических наук; зав. кафедрой «Компьютерные системы» Ферганского филиала Ташкентского университета информационных технологий им. Мухаммада Ал-Хоразмий. Фергана, Республика Узбекистан. ORCID: https://orcid.org/0000-0002-9471-4893. E-mail: mamatiso 2015@yandex.ru

Рахимов Рустам Хакимович, доктор технических наук, профессор; зав. 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