YUPQA TERMOMAGNITELASTIK PLASTINANING (GEOMETRIK NOCHIZIKLI) JARAYONINI MATEMATIK MODELINI QURISH

THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 167 YUPQA TERMOMAGNITELASTIK PLASTINANING (GEOMETRIK NOCHIZIKLI) JARAYONINI MATEMATIK MODELINI QURISH F.M.Nuraliyev1, O.K.Abdullayev2 1,2Toshkent axborot texnologiyalari universiteti, Oʻzbekiston https://doi.org/10.5281/zenodo.17740149 Ushbu maqolada Gamilton-Ostrogradskiy variatsiyali tamoyili asosida murakkab shaklga ega yupqa plastinaning termo-elektro-magnit-elastiklik deformatsiyalanish jarayonining chiziqli matematik modeli keltirib chiqarilgan[2,3]. Plastinaning harakat tenglamasini ishlab chiqishda koʻchishning oʻzgarish qonunlari sifatida Kirxgofa-Lyava gipotezasidan foydalanamiz [3,4]. 1 2 3 , , ; ww u u z u v z u w xy  = − = − =  (1) bu yerda: ,,u v w − koʻchishlar. Kirxgof-Lyav gipotezasiga koʻra, yupqa plastina qalinligi boʻylab deformatsiya boʻlmaydi, natijada masalaning uch oʻlchovli modeli ikki oʻlchovli matematik modelga aylanadi. Gamilton-Ostrogradskiy variatsion tamoyilining umumiy koʻrinishi [2]. 0 t (δK δΠ +δA)dt = ;−  (2) bu yerda: δ− variatsiya, K – kinetik energiya, P – potensial energiya, A – tashqi hajm va sirt quchlari bajargan ish. Kinetik energiyaning oʻzgarishini hisoblashda quyidagi munosabatdan foydalanamiz: 33 1 1 2 2 ; t t V uu u u u u Кdt dVdt t t t t t t              = + +           (3) gde: ρ – qaralayotgan obekt materialining zichligi; 1 2 3 ,,u u u − koʻchish, V− hajmi, t− vaqt. Bu yerda 1 2 3 ,,u u u larning oʻrniga (1) formuladagi qiymatlarini keltirib qoʻyamiz. Variatsiya ostidagi kinetik energiyani boʻlaklarga boʻlib integrallaymiz. Bu yerda plastina qalinligi boʻyicha birlashtiriladi. ( ) 3 2 3 2 2 2 2 3 4 3 4 2 2 2 2 2 2 2 12 12 . 12 4 12 yt x y y x x tt t x y u v w h w h w K h u h v h w dydx w dy w dx t t t t x t y u v w h w h w h u h v h w w w dydxdt t t t t x t y                            = + + + + −                − + + + +              bunda quyidagi : 3 4 3 4 2 2 2 2 , 12 12 h W h W x t y t       hadlarning tasir qiymat juda kichikligi sabab tenglamadan tushirib qoldiriladi. Natijada variatsion kinetik energiyaning umumiy THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 168 koʻrinishi hosil boʻldi: 2 2 2 2 2 2 . x y t x y t u v w u v w K h u h v h w dydx h u h v h w dydxdt t t t t t t                      = − + − − +               (5) Potensial energiya variatsiyani aniqlashning umumiy koʻrinishi [5]: ; xx xx yy yy xy xy v e e e dv          = + +   (6) Bu yerda e− deformatsiya,  − kuchlanish Koshi munosabatlariga koʻra Kirxgofa-Lyava (1) gipotezasidan foydalanib deformatsiya aniqlanadi va Koshi munosabatlariga koʻra variatsiyalaymiz [5]. Variatsiya potensiali (6) - formuladagi ,, xx yy xy e e e ni (8) – formuladan keltirib qoʻyiladi. Bu yerda plastina qalinligi boʻyicha birlashtiriladi. ( ) ( ) 0 0 1( ), 1( ), 1, xx xx yy T yy yy xx T xy xy e T T E e T T E eE         = − + −    = − + −   + =  (9) ( ) ( ) 0 2 0 2 1( ), 1 1 1( ), 1 1 . 1 xx xx yy T yy yy xx T xy xy Ee e T T Ee e T T Ee             = − − −  − −  = − − −  − −  = + (10) Potensial energiya variatsiyasi (1) - formuladagiga ,, xx yy xy e e e ni (3) – formuladan keltirib qoʻyiladi. Bu yerda plastina qalinligi boʻyicha birlashtiriladi. Dyuamel-Neyman munosabati va Guk qonunidan foydalanib ushbu ifodani keltirib chiqaramiz[1]. 2 1 2 2 2 2 2 12 2 1 2 2 2 2 1 2 1(7) 2 2, 1 2 1 2 xx yy xy xx yy uu w w w ez x x x x x uv w w w ez y y y y y uuu v w w w ez y x y x x y x y uu w w w ez x x x x x uv w w w ez y y y y y                = = − +           = = − +            = + = + − +              = = − +          = = − +      2 12 (8) 2, xy uuu v w w w ez y x y x x y x y                  = + = + − +          THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 169 ( ) 22 0 2 2 2 22 0 2 2 2 2 1 ( ) , 11 1 ( ) , 11 2. 21 xx T yy T xy E u w v w z z T T x x y y E v w u w z z T T y y x x E u v w z y x x y                   = − − + − −   −     −          = − − + − −   −     −         = + −  +        (11) E – elastik modul,  – Puasson koeffisenti, T – issiqlikka chidamlilik koeffitsienti. 0 T – plastinkaning dastlabki harorati, T – plastinka harorati Bu yerda plastina qalinligi boʻyicha birlashtiriladi, va matematik modeli hosil qilinadi. 2 2 2 2 2 2 2 2 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ()11 1 1 2 2(1 ) 2 0 1 T x x x x zx a T T u Eh u v w w Eh u v w w w w ht x x x x x y x y y x y x y R N q v Eh v u ht y x                 −               − + − + − + + +  +  +            −      − +                    + + + + + =     − + − +  −   2 2 2 2 0 2 2 2 2 3 4 4 4 3 2 2 4 2 2 4 ()11 1 2 2(1 ) 2 0 11 2 12 1 12 T y y y y zy a T T w w Eh v u w w w w y y y x x y x y x x y R N q w Eh w w w Eh ht x x y y              −           − + + +  +  +              − −                    + + + + + =       − −  + + −     −       4 22 22 22 (1 ) 0 1 z z z z zz T w xy R N q T T T x y a t                  + −   + + + + + =        +=      FOYDALANILGAN ADABIYOTLAR 1. 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