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Двусторонняя аналитическая оценка первой частоты плоской фермы

Двусторонняя аналитическая оценка первой частоты плоской фермы

Автор: Воробьев Олег

Журнал: Строительство уникальных зданий и сооружений @unistroy

Статья в выпуске: 7 (92), 2020 года.

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Объект исследования - статически детерминированная консольная ферма. Трасса состоит из прямоугольных панелей с направленными вниз диагональными балками. Ферма имеет две опоры, одна из которых неподвижно-шарнирная, а другая роликовая. Массы располагаются в узлах верхней и нижней поясов. Силы в стержнях и реакции на опорах определяются методом совместной изоляции. Вертикальное смещение узлов выводится из метода Максвелла-Мора с предпосылкой линейной упругости. Зависимость вертикального смещения, оценок Дункерли и Рэлея частоты первичной фермы от количества панелей выводится из индуктивного анализа набора конкретных ферм с увеличивающимся количеством панелей. Рекуррентные уравнения, отвечающие определенным коэффициентам, выводятся с использованием специальных функций системы компьютерной алгебры Maple. Полученные решения являются полиномиальными с количеством панелей в качестве переменных. Коэффициент Рэлея рассчитывается с предположением, что первая мода вибрации равна прогибу фермы под действием равномерно распределенной нагрузки. Построены графики зависимости полученных оценок от масс узлов, количества панелей, жесткости и размеров фермы.

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Ферма, аналитическое решение, частота, метод Дункерли, фактор Рэлея, клен, символьная индукция

Короткий адрес: https://sciup.org/143172555

IDR: 143172555   |   УДК: 69   |   DOI: 10.18720/CUBS.92.4

Bilateral analytical estimation of the first frequency of a plane truss

The object of research is the statically determinate cantilever truss. The trass consists of rectangular panels with downward diagonal beams. The truss has two supports, one of which is fixed hinged, and another one is roller support. Masses are located in the nodes of top and bottom chords. Forces in the bars and reactions at supports are determined using the method of joint isolation. The vertical displacement of nodes is derived from the Maxwell-Mohr method with the premise of linear elasticity. Dependence of vertical displacement, Dunkerley’s and Rayleigh’s estimations of primary truss frequency on the number of panels is deduced from the inductive analysis of the set of particular trusses with an increasing number of panels. Recurrence equations that meet particular coefficients are derived using special functions of the computer algebra system Maple. Obtained solutions are polynomial, withthe number of panels as variables. Rayleigh’s quotient is calculated with the assumption that the first mode of vibration is equal to truss deflection under the uniformly distributed load. Graphs of the dependencies of obtained estimations on nodes masses, the number of panels, stiffness, and size of the truss are plotted.

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