ilmsarchashmalari@umail.uz +998 91-915-45-55 220100, Urganch shahri, Hamid Olimjon koʻchasi, 14-uy.
O'zbekcha Русский English
Arxiv / №9 · 2026 / Maqola
01.00.00 – FIZIKA-MATEMATIKA FANLARI

GARMONIK YUKLAMA OSTIDAGI 3-TARTIBLI INERTIO-VISKOELASTIK MODELNING ANALITIK YECHIMI TAHLILI

Annotatsiya
O'ZBEKCHA

Garmonik yuklama ta’siridagi materiallarning tebranma dinamikasini tavsiflovchi 3-tartibli inertio-viskoelastik reologik model tadqiq qilindi. Ishning maqsadi uchinchi tartibli differensial tenglamaning analitik yechimini keltirib chiqarishdan iborat. Integrallash o‘zgarmaslari hamda tebranish amplitudalari Kramer qoidasi bo‘yicha aniqlanib, deformatsiyalanish kinetikasi so‘nuvchi va turg‘un teb-ranishlar superpozitsiyasi ko‘rinishida yoritildi. Taklif etilgan matematik apparat va Python dasturi reo-logik murakkab materiallarning dinamik xossalarini bashorat qilishga xizmat qiladi.

Kalit so'zlar: inertio-viskoelastiklik, 3-tartibli reologik model, analitik yechim.
РУССКИЙ

Исследована инертио-вискоэластичная реологическая модель 3-го порядка, опи-сывающая колебательную динамику материалов при гармоническом нагружении. Цель работы - выведение аналитического решения дифференциального уравнения третьего порядка. Постоянные интегрирования и коэффициенты амплитуды колебаний определены по правилу Крамера, а кине-тика деформирования представлена в виде суперпозиции затухающих и установившихся колеба-ний. Предложенный математический аппарат и программный комплекс на Python служат для прогнозирования динамических свойств реологически сложных материалов.

Ключевые слова: инертио-вискоэластичность, реологическая модель 3-го порядка, аналити-ческое решение.
ENGLISH

A 3rd-order inertio-viscoelastic rheological model describing the vibrational dynamics of materials under harmonic loading was investigated. The study aims to derive an analytical solution for the third-order differential equation. Integration constants and vibration amplitude coefficients were determined via Cramer's rule, representing deformation kinetics as a superposition of damped and steady-state oscillations. The proposed mathematical framework and Python software package provide a tool for predicting the dynamic properties of rheologically complex materials.

Keywords: inertio-viscoelasticity, 3rd-order rheological model, analytical solution.
Maqola · PDF
PDF yuklab olish Yangi oynada ochish

Foydalanilgan adabiyotlar

  1. Ferry J.D. (1980). Viscoelastic properties of polymers (3rd ed.). John Wiley & Sons. https://doi.org/ 10. 1002/pol.1981.130191012
  2. Wineman A. (2009). Nonlinear viscoelastic solids–A review. Mathematics and Mechanics of Solids, 14(3), 300–366. https://doi.org/10.1177/1081286509103896
  3. Mainardi F., Spada G. (2011). Creep and relaxation in linear viscoelasticity: Is there always a duality? Rheologica Acta, 50(1), 15–30. https://doi.org/10.1007/s00397-010-0494-6
  4. Christensen R.M. (2010). Theory of viscoelasticity: An introduction (2nd ed.). Dover Publications. https:// doi.org/10.1016/C2009-0-21932-1
  5. Safarov I.I., Teshaev M.K., Akhmedov A.B. (2020). Numerical study of dynamic characteristics of viscoelastic systems. Journal of Applied Mathematics and Computational Mechanics, 19(2), p. 75–86. https://doi.org/ 10.17512/jamcm.2020.2.07
  6. Khudoynazarov K., Abdikarimov A. (2019). Torsion vibrations of a viscoelastic cylindrical shell filled with a liquid. AIP Conference Proceedings, 2186(1), Article 080005. https://doi.org/10.1063/1.5138012
  7. Sobotka Z. (1984). Rheology of materials and engineering structures. Elsevier. https://doi.org/10.1016/ C2009-0-09319-3
  8. Barnes H.A., Hutton J.F., Walters K. (1989). An introduction to rheology. Elsevier. https://doi.org/ 10.1016 /C2009-0-08805-X
  9. Tschoegl N.W. (1989). The phenomenological theory of linear viscoelastic behavior: An introduction. Springer-Verlag. https://doi.org/10.1007/978-3-642-73602-5
← Songa qaytish