FAILURE OF THIN MAGNETELASTIC PLATES OF COMPLEX CONSTRUCTION MATHEMATICAL MODELING OF DEFORMATION PROCESSES Текст научной статьи по специальности «Компьютерные и информационные науки»

MURAKKAB KONSTRUKTIV SHAKLDAGI YUPQA MAGNITELASTIK PLASTINALARNING NOCHIZIQLI DEFORMATSIYALANISH JARAYONLARINI MATEMATIK MODELLASHTIRISH

dotsent, Shohruh SAFAROV ALFRAGANUS UNIVERSITY ORCID: 0000-0001-7816-4409 Abduaziz ABDURAXMANOV

ALFRAGANUS UNIVERSITY

Annotatsiya. Maqola Gamilton-Ostrogradskiy variatsion tamoyiliga asoslanib murakkab konstrukti'v shakldagi yupqa magnitelastik plasti'nalarning nochiziqli geometrik deformatsiyalanish jarayonlarini matemati'k modeli ishlab chiqishga bag'ishlanadi. Bunda Kirxgof-Lyav gipotezasidan foydalanib uch o'lchovli matemati'k model ikki o'lchamli ko'rinishga o'tkazildi. Potensial va Kinetik energiyaning variatsion ko'rinishlari hamda tashqi kuchlar bajargan ishning variatsion korinishi topildi. Bular Koshi munosabatlari, Guk qonuni va Lorens kuchi hamda Maksvell elektromagnit tenzor ko'rinishidan foydalanib aniqlandi. Magnitelastik plastinaning deformatsion kuchlanish holati'ga elektromagnit maydon ta'sirlari ko'rildi. Natijada ko'chishga nibatan boshlang'ich va chegaraviy shartlarga ega bo'lgan, xususiy hosilali differensial tenglamalar sistemasi ko'rinishidagi matemati'k model ishlab chiqildi. Masalani yechish uchun algebra mantiqiy R-funksiya, Bubnov-Galerkin, Nyumark, Gauss, Gauss kvadratlar hamda Iterasiya sonli usullaridan foydalanib hisoblash algoritmi ishlab chiqildi. Tadqiqotni hisoblash tajribalarini o'kazish uchun amaliy dasturiy majmua yaratildi. Olib borilgan hisoblash tajribalarida magnit elastik plastinaning turli mexanik holatlari, chegaralari qattiq mahkamlangan, bir tomoni sharnir ikkinchi tomoni erkin holatida hisoblash tajribalari o'tkazilib sonli natijalar olindi. O'takazilgan hisoblash natijalarining qiyosiy tahlillari maqolada keltirilgan.

Kalit so'zlar: Gamilton-Ostrogradskiy tamoyili, Bubnov Galerkin, Koshi munosabatlari, Guk qonuni, Maksvell elektromagnit tenzori, murakkab konfiguratsiyali magnitelastik yupqa plastina R-funksiya.

Abstract. The article is devoted to the development of a mathematical model of nonlinear geometric deformation processes of thin magnetoelasti'c plates of complex structural form based on the Hamilton-Ostrogradsky variational principle. In this case, the three-dimensional mathematical model was transferred to a two-dimensional view using the Kirchhoff-Liav hypothesis. Variations of potential and kinetic energy and variation of work done by external forces were found. These are determined using Cauchy's relation, Hooke's law and Lawrence force, and Maxwell's electromagnetic tensor representation. Effects of the electromagnetic field on the deformation stress state of the magnetoelasti'c plate were observed. As a result, a mathematical model in the form of a system of partial differential equations with initial and boundary conditions for displacement was developed. To solve the problem, a calculation algorithm was developed using algebraic logic R-function, Bubnov-Galerkin, Newmark, Gauss, Gaussian squares and Iteration numerical methods. A practical software package has been created to conduct research computing experiments. In the conducted calculation experiments, various mechanical states of the magneto-elastic plate, the limits of which are fixed, one side is hinged and the other side is free, the calculation experiments were conducted and numerous results were obtained. Comparative analysis of the results of the calculation is presented in the article.

Key words: Hamilton-Ostrogradsky principle, Bubnov Galerkin, Cauchy relation, Hooke's law, Maxwell's electromagnetic tensor, complex configuration magnetoelasti'c thin plate R-function

Kirish

Bugungi kunda elektromagnit maydonlarning elektr o'tkazuvchanlik va magnitelastiklikning nochiziqli nazariyalari, xususan ikki yoki undan ortiq maydonlarning o'zaro bog'liqlik nazariyasiga asoslangan ilmiy tadqiqot ishlari izchil suratlarda rivojlanmoqda. Yupqa magnitelastik konstrukti'v elementlar mashinasozlik, samolyotsozlik, kemasozlik va inshootlar qurilishi obektlarining muhim tarkibiy elementlarini tashkil qiladi. Dunyoning rivojlangan davlatlari, jumladan, AQSh, Xitoy, Germaniya, Rossiya Federatsiyasi, Yaponiya, Eron va boshqa mamlakatlarda elektro'tkazuvchan murakkab konfiguratsiyali yupqa plastina va qobiq shaklidagi konstrukti'v mikroelementlarning elektromagnit maydonda magnitelastiklik o'zaro ta'siri muammolari muhim ahamiyat kasb etmoqda.

Jahon miqyosida nochiziqli qonuniyatlar asosida murakkab konfiguratsiyali yupqa plastina va qobiqlar shaklidagi konstruktiv elementlarning keng ravishda ishlab chiqarilishi va qo'llanilishi dolzarb hisoblanadi. Elektromagnit maydonlarning yupqa elektr-o'tkazuvchan jismlarning deformatsiyalanish holati'ga ta'sir etish jarayonlarini modellashtirish, murakkab konfiguratsiyali magnitelastik plastina va qobiqlarning asosiy chegaraviy shartlarini qanoatlantiruvchi yechimlar tuzilmasini R-funksiya usulida modellashtirish algoritmlari va dasturiy vositalarning yangi avlodini ishlab chiqish alohida kasb eti'b bormoqda.

Jahonda va yuztimizda yupqa elektro'tkazuvchan jismlarning magnitelastikligini tadqiq qilish masalalari bo'yicha bir qator olimlar: D.I.Bardzokas, S.A.Ambarsumyan, G.Ye.Bagdasaryan, M.V.Belubekyan X.A.Raxmatulin, V.K.Kobulov, B.Kurmanbaev, Sh.A.Nazirov, T.Yuldashev, A.A.Xoljigitov, R.Sh.Indiaminov, F.M.Nuraliev kabi mamlakatimiz olimlari ilmiy-tadqiqot ishlarini olib borishgan.

Adabiyotlar tahlili shuni ko'rsatadiki, elektromagnit maydon ta'siridagi elektr o'tkazuvchan murakkab konstruktiv shakldagi magnitelastik yupqa plasti'nalarning geometrik nochiziqli deformatsiyalanish jarayonlarini matematik modellashtirish muammolari hozirgi kunda yetarli darajada o'rganilmagan.

Matematik model ishlab chiqish

Magnitelastik plasti'nanig gometrik nochiziqli deformatsiyalanish jarayonining matematik modelini ishlab chiqish uchun Gamilton-Ostrogradskiy variatsion tamoyili, Kirxgof-Lyav gipotezasi, Koshi munosabatlari, Guk qonuni Lorens kuchi hamda Maksvell elektromagnit tenzor ko'rinishidan foydalanildi.

-ph 8U + ^ + ^ + N + R, + q, + T. = 0,

8t2 8x

dy

82v 8NW 8Nxy -Ph ^ + + ~TL + Ny + Ry + qy + = 0

-ph

8t2 8y

82w 8 2M - + -

8x

8t2

8x2

■ + 2-

8 2M 82M - + -

8x8y 8y 2

+Nx

82 w

"8X7

+N

82w

82w

+ N

yy 8y2 xy 8x8y

+

+

(8N,x 8N,y W (8N„ 8N„ \

+

8x 8y j

8x

+

+

8y 8x j 8y

8w

— + N + R + q + T = 0.

^ z z iz zz

Boshlag'ich shartlar va chegaraviy shartlar:

ph — Su 8t

8v

= 0, ph — Sv

8t

= 0, ph8wSw 8t

= 0,

(N + NP + NT)Su| = 0, (N + NP + NT )Sv| = 0,

\ xx Px ix/ |x ' \ xy Py Txy / '

MxxS

8w 8x

= 0, MS —

xy 8y

8w

= 0, MWS —

yy 8y

8w

= 0, MS —

xy 8x

= 0,

(Nyy + NFr + Niyy)Sv| = 0, (Nxy + Nfx + Niyx)Su| = 0,

8w 8w 8Mxx 8Mxy Nxx + Nxy T---8xxx -+ NPz + NT

8x 8y 8x 8y

Sw

= 0,

N 8w + N 8w-8Mxy + N + N 8y + Nxy 8x 8y 8x + +

Sw

= 0.

N , N , N

6y epga xx yy xy - n^acrurnaHHHr Ka.nnH.nMrM 6yMMna HopMa.n Ba ypyHMa Kyn.napM.

M ,M ,M p-

xx yy xy - n^aciMHaHMHr эгм.пмw Ba 6ypa.nMw MOMemTnapM, y wmcm 3MH.nMrM, h- n.nacrMHa

x

y

x

y

x

y

^линлиги,

Rx, Ry, Rz, Nx, Nv, Nz -

сирт кyчлapи,

y хосил 6УЛУВЧИ xaжм кyчлapи, x y

Г i 7 ГТ1 ft 7 ГШ 7 ra i ra i

xx9 xy9 xz9 yy9 yz9 zx9

qx 9 qy 9 qz 9 Tzx , Tzy , Tzz -

хосил 6улувчи контyp кyчлapи.

Масалани сонли ечишнинг х,исоблаш алгоритми

1-rasm. Tenglamani echishning hisoblash algoritmi. Hisoblash tajribalari tahlílí

Xarakat tenglamasidagi (1) noma'lumlarni topish uchun, Bubnov - Galerkin variatsion usuli, Gauss kvadratlar, Gauss, Nyumark, hamda Iteratsiya sonli usullarini birgalikda qo'llab, yupqa plastinaning OZ

o'qi bo'ylab ko'chishi U(x'y),V(x'yW(x'y).

Dastlab murakkab konfiguratsiyali sohaning analitik tenglamasini R-funksiya usuli bilan quriladi. Hisoblash tajribalari murakkab konsiruksion shakldagi yupqa plastina (1-rasm) uchun mexanik va geometrik parametrlar hisobga olinib o'tkaziladi:

a = 1, b = 1, h = 0.01, r = 0.2, v = 0.3, t0 = 0, q = 1, Hx = Hy = Hz = 10кЭ, Qz = 1.

M

t/ l( >л >A\

Vv )i Ji //

2-rasm. Murakkab shakldagi plastina. Murakkab sohaning R-funksiya orqali olingan analitík tenglamasi quyidagicha ifodalanadi.

/г

(a2 - x2 ) (b2 - y2 ) (x2 + y2 - r2 )

V f> 0, /2 = V 0, /3 = *-1-'-> 0,

2b

2r

2a

® = (( /1л /2 )л УЗ ) ;

bu erda/1,/2,/з - sohani xarakterlaydigan funksiyalar. œ- soha.

Tenglamani echishda iteratsiya usuli qo'llanilib plastina o'rta sirtining ui(x' vi(x' Wi(x'y qiymatlari topiladi. Kirxgof gipotezasiga ko'ra yupqa plastina o'rta sirtining ko'chishlari

ui (x>y)->vi(x> y) Ox va Oy o'qiga nisbatan kichik bo'lganligi uchun tajribalarda asosan Wi(x'y)

nati'jalari tahlil qilinadi. Murakkab shakldagi yupqa plastina (2-rasm) x[-1:1] y=0 da Wi(x'y) ning taqribiy sonli qiymatlari olindi. Bunda murakkab konstruktiv shakldagi magnitelastik plasti'naning chegaralari sharnir mahkamlangan shartlar bilan qo'yilgan masalaning hisoblash tajribalari o'tkazilgan sonli nati'jalari (1-jadval) olingan hamda egilish grafik tasviri (3-rasm) olingan.

1-jadval.

x Mexanik kuchlar Mexanik va Magnit kuchlar

-1 0 0

-0.9 0.0004 0.0005

-0.8 0.0009 0.0011

-0.7 0.0013 0.0016

-0.6 0.0014 0.0018

-0.5 0.0013 0.0016

-0.4 0.0010 0.0012

-0.3 0.0006 0.0007

-0.2 0 0

0.2 0 0

0.3 0.0006 0.0007

0.4 0.0010 0.0012

0.5 0.0013 0.0016

0.6 0.0014 0.0018

0.7 0.0013 0.0016

0.8 0.0009 0.0011

0.9 0.0004 0.0005

1 0 0

w(x,y,t), x[-1;1] y=x, t=0.5

-0,002 i 0,0015 - /' m t w\ n nni - _^

Jß v\

^^^ u,uu± 0,0005 -

/ \

-1 -0,5 0,5 1

■■Mex -Mex+Mag

3-rasm. Tashqi kuchlar ta'sirida murakkab shaklli yupqa plasti'naning OZ o'qi bo'ylay egilishi. O'tkazilgan tajriba natijasi shuni ko'rsatadiki elekr o'tkazuvchan magnitelastik plastina magnit maydon kuch ta'sirida deformatsiyalanishi (OZ o'qi bo'ylab egilishi) 18 % ga oshishi kuzati'ldi.

2-jadval.

x y Elektromagnit maydon kuchlar ta'siri qaralmagan wi(x y) Elektromagnit maydon kuchlar ta'sir qilganda

-1 О О О

О.9 О О.ООО1 О.ООО1

О.8 О О.ООО6 О.ООО7

О.7 О О.ОО16 О.ОО2О

О.6 О О.ОО27 О.ОО34

О^ О О.ОО32 О.ОО4О

О.4 О О.ОО29 О.ОО37

О.3 О О.ОО19 О.ОО24

О.2 О О О

О.2 О О О

О.3 О О.ОО19 О.ОО24

О.4 О О.ОО29 О.ОО37

0.S О О.ОО32 О.ОО4О

О.6 О О.ОО27 О.ОО34

О.7 О О.ОО16 О.ОО2О

О.8 О О.ООО6 О.ООО7

О.9 О О.ООО1 О.ООО1

1 О О О

w(x,y), x[-1;1], y=O,t=0.5

0,005 0,004 0,003 0,002 0,001

-0,8 -0,6 -0,4

-0-0,2 0

--0,001 J

X

0,2 0,4 0,6 0,8

► — Мех

Мех+Mag

1

1

Yupqa plastinaning chegarasi bir tomoni erkin, qolgan tomonlari va markazi qattiq mahkamlangan shartlar bilan olingan sonli natijalar grafik tasviri (S-rasmda) uchun analitik tenglama qurildi va bu quyidagi formula orqali ifodalandi.

Hisoblash tajribasidagi geometrik va mexanik parametrlar quyidagilarni tashkil etadi.

3-jadval

Plasti'naning chegaralari aralash shartlarda olingan sonli natijalar

Mexanik kuchlarni hisobga olganda w(x,y,t) у=О t=0.1

Mexanik va Magnit maydon kuchlarini

x

hisobga olganda w(x,y,t) y=0 t =0.1

-1 0.0116 0.141

-0.9 0.073 0.0.085

-0.8 0.046 0.051

-0.7 0.028 0.0030

-0.6 0.017 0.0018

-0.5 0.010 0.0011

-0.4 0.006 0.0006

-0.3 0.003 0.003

-0.2 0.001 0.001

-0.1 0 0

0.1 0 0

0.2 0.001 0.002

0.3 0.004 0.004

0.4 0.008 0.008

0.5 0.012 0.012

0.6 0.015 0.017

0.7 0.017 0.020

0.8 0.013 0.016

0.9 0.006 0.007

1 0 0

w(x,y,t)x[-1:1], y=0,t=0.5

-0,160 i 0,140 -0,120 -0,100 0,080 -n ncn -

0,060 0,040 - ^ r* non -

0,020

-1

-0,5

Mex

Mex+Mag

0,5

5-rasm. Plasti'naning chegaralari qattiq mahkamlangan, markazi erkin holatda olingan egilish grafigi.

Xulosa

Murakkab konfiguratsiyali magnitelasti'k yupqa plasti'naning geometrik nochiziqli deformatsiyalanish jarayonlarining matemati'k modeli qurildi. Uni echish uchun hisoblash algoritmi tuzildi. Tahliliy tajribalar o'tkazish uchun dasturiy vosita yaratildi. Murakkab konstruktiv shakldagi magnitelasti'k yupqa plastinalarning elektromagnit kuchlar ta'sirida geometrik nochiziqli deformatsiyalanish jarayonlari turli chegaraviy sharlar asosida o'rganildi va sonli natijalar olindi hamda qiyosiy tahlillar keltirildi. O'tkazilgan tajriba nati'jalari shuni ko'rsatadiki magnit maydon kuchlarning yupqa magnitelasti'k plasti'nalarga ta'siri kichik bo'lsada mavjudligi aniqlandi va bu plasti'naning deformatsiyalanish jarayoniga bevosita ta'siri isbotlandi.

Foydalanilgan adabiyotlar Kabulov V.K. Algorithmization in the theory of elasticity and deformation theory of plasticity Tashkent Science 1966. 392 p.

1

Ambartsumyan S.A., Belubekyan M.V. Some problems of electromagneto elasticity of plates. - Erevan: 1991.-144 p.

Kurpa L.V. Metodom R-funksi dlya resheniya lineynbix zadach izgiba i kolebaniy pologix obolochek. Xarkov NTU XPI 2009. 391s.

Leybenzon L.S. Kurs teorii uprugosti. Moskva 1977. 272s. (17-str).

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https://ieeexplore.ieee.org/document/9011903

F. Nuraliev, S. Safarov and M. Arti'kbayev, "A computational algorithm for calculating the effect of the electromagnetic fields to thin complex configured plates," 2020 International Conference on Information Science and Communications Technologies (ICISCT), 2020, pp. 1-4, https://ieeexplore.ieee.org/document/9351447

F. Nuraliev, S. Safarov and M. Arti'kbayev, "Solving the problem of geometrical nonlinear deformation of electro-magnetic thin plate with complex configuration and analysis of results," 2021 International Conference on Information Science and Communications Technologies (ICISCT), 2021, pp. 01-05, https://ieeexplore.ieee.org/document/9670282

Nuraliev F.M., Aytmuratov B.Sh., Safarov Sh.Sh., Arti'kbayev M.A. Mathematical modeling of geometric nonlinear processes of electromagnetic elastic thin plates of complex configuration // Scientific journal «Problems of Computational and Applied Mathematics» № 1(38), 2022. https://scholar.google.com/citations?view op=view citation&hl=en&user=ialkwscAAAAJ&citation f or view=jalkwscAAAAJ:9yKSN-GCB0IC

F. Nuraliev, S. Safarov, M. Arti'kbayev and A. O.Sh, "Calculation Results of the Task of Geometric Nonlinear Deformation of Electro-magneto-elasti'c Thin Plates in a Complex Configuration," 2022 International Conference on Information Science and Communications Technologies (ICISCT), Tashkent, Uzbekistan, 2022, pp. 1-4,. https://ieeexplore.ieee.org/document/10146920