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Arxiv / №2 · 2026 / Maqola
01.00.00 – FIZIKA-MATEMATIKA FANLARI

ERKINLIK DARAJASI CHEKLI BO‘LGAN KO‘P MASSALI QOVUSHQOQ-ELASTIK SISTEMALARNING ERKIN VA MAJBURIY TEBRANISHLARI

Annotatsiya
O'ZBEKCHA

Ushbu maqolada erkinlik darajasi chekli bo‘lgan ko‘p massali qovushqoq-elastik sis-temalarning erkin va majburiy tebranishlarini o‘rganamiz. Misol sifatida murakkab ko‘p massali tebranish sistemalari ko‘rib chiqilgan. Ikkinchi tur Lagranj tenglamalari asosida integro-differensial tenglamalar sistemasi hosil qilingan. Mexanik sistemalarning erkin va majburiy tebranishlarini hisoblash uchun Myul-ler va Gauss usullariga asoslangan hisoblash metodikasi keltirilgan. Sonli natijalar olinib, tahlil qilingan.

Kalit so'zlar: erkin tebranishlar, erkinlik darajasi chekli bo‘lgan qovushqoq-elastik sistema, Lagranj tenglamasi, qattiqlik operatori.
РУССКИЙ

В работе рассматриваются собственные и вынужденные колебания много- массовых вязкоупругих систем с конечным числом степеней свободы. В качестве примера рассма-триваются сложные много-массовые колебательные системы "автомобиль – амортизируемое из-делие". На основе уравнений Лагранжа второго рода получена системы интегро-дифференциаль-ных уравнений. Приведена методика расчета собственных и вынужденных колебаний механичес-ких систем на основе метода Мюллера и Гаусса. Полечены численные результаты и их анализ.

Ключевые слова: собственные колебания, вязкоупругая система с конечным числом степеней свободы, уравнение Лагранжа, операторные жесткости.
ENGLISH

This paper investigates the free and forced vibrations of multi-mass viscoelastic systems with a finite number of degrees of freedom. As an illustrative example, complex multi-mass vibratory systems of the type “vehicle – damped (shock-isolated) payload” are considered. Based on the Lagrange equations of the second kind, a system of integro-differential equations is derived. A computational methodology for determining the free and forced vibrations of mechanical systems using the Müller and Gauss methods is presented. Numerical results are obtained and their analysis is provided.

Keywords: free vibrations, viscoelastic system with a finite number of degrees of freedom, Lagrange equations, operator stiffnesses.
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