Structural Mechanics of Engineering Constructions and BuildingsStructural Mechanics of Engineering Constructions and BuildingsСтроительная механика инженерных конструкций и сооружений1815-52352587-8700Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)4521710.22363/1815-5235-2025-21-2-108-117NJOXALAnalytical and numerical methods of analysis of structuresАналитические и численные методы расчета конструкцийResearch ArticleNatural Frequency Spectrum and Fundamental Frequency Formula for Plane Periodic Lattice TrussСпектр собственных частот и формула для основной частоты плоской регулярной фермы решетчатого типаhttps://orcid.org/0000-0002-8588-38718679-6853KirsanovMikhail N.КирсановМихаил Николаевич

Doctor of Physical and Mathematical Sciences, Professor of the Department of Robotics, Mechatronics, Dynamics and Strength of Machines

доктор физико-математических наук, профессор кафедры робототехники, мехатроники, динамики и прочности машин

c216@ya.ruNational Research University “MPEI”Национальный исследовательский университет «Московский энергетический институт»1107202521210811723072025Copyright ©; 2025, Kirsanov M.N.Copyright ©; 2025, Кирсанов М.Н.2025Kirsanov M.N.Кирсанов М.Н.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/structural-mechanics/article/view/45217

The goal is to determine the free vibration natural frequency spectrum for a plane statically determinate truss with a cross-shaped lattice. The truss members are elastic and have the same stiffness. Both truss supports are pinned; the truss is externally statically indeterminate. A model, in which the mass of the structure is uniformly distributed over its nodes, and their vibrations occur vertically, is considered. The Maxwell-Mohr method is used to determine the stiffness of the truss. The member forces included in the formula are calculated by the method of joints using the standard operators of Maple mathematical software in symbolic form. The eigenvalues of the matrix for trusses with different numbers of panels are determined using the Maple system operators. Spectral constants are found in the overall picture of the frequency distribution constructed for trusses of different orders. A formula for the relationship between the first frequency and the number of panels is derived from the analysis of the series of analytical solutions for trusses of different orders. A simplified version of the Dunkerley method is used for the solution, which gives a more accurate approximation in a simple form. The relationship between the truss deflection under distributed load and the number of panels was found. Spectral constants were found in the frequency spectrum.

Для плоской статически определимой фермы с крестообразной решеткой определяется спектр собственных частот свободных колебаний. Стержни фермы упругие и имеют одинаковую жесткость. Обе опоры фермы неподвижные шарниры, ферма внешне статически неопределима. Рассмотрена модель, в которой масса конструкции равномерно распределена по ее узлам, а их колебания происходят по вертикали. Для определения жесткости фермы применен метод Максвелла - Мора. Усилия в стержнях, входящие в формулу, рассчитывались методом вырезания узлов с применением стандартных операторов системы компьютерной математики Maple в символьной форме. Собственные числа матрицы для ферм с различным числом панелей разыскиваются с помощью операторов системы Maple. В общей картине распределения частот, построенной для ферм различного порядка, обнаружены спектральные константы. Из анализа последовательности аналитических решений для ферм разного порядка выведена формула зависимости первой частоты от числа панелей. Для решения использован упрощенный вариант метода Донкерлея, дающий более точное приближение в простой форме. Найдена зависимость прогиба фермы под действием распределенной нагрузки от числа панелей. В спектре частот обнаружены спектральные константы. Выведена формула зависимости прогиба фермы от числа панелей.

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