Unsteady thermoelastic diffusion vibrations of the Bernoulli-Euler beam under the action of a distributed transverse load

Author details

Andrei V. Zemskov
Moscow Aviation Institute (National Research University), Moscow, Russian Federation; Research Institute of Mechanics Lomonosov Moscow State University, Moscow, Russian Federation
Hao Le Van
Moscow Aviation Institute (National Research University), Moscow, Russian Federation

https://doi.org/10.15593/perm.mech/2023.3.07

The paper deals with the problem of unsteady vibrations of the Bernoulli – Euler beam, taking into account relaxation of temperature and diffusion processes. The original mathematical model includes a system of equations of non-stationary bending oscillations of the beam taking into account heat and mass transfer, which is obtained from the general model of thermoelastic diffusion for continuum using variational D'Alembert principle. Based on the obtained equations, the statement of the initial-boundary problem concerning bending of the hinged orthotropic beam, which is under the action of thermo- elastic diffusion perturbations distributed on the surface, is formulated. Solutions of the problem of unsteady thermoelastic diffusion vibrations of the beam are sought in integral form. The kernels of integral representations are Green's functions, for finding of which decompositions into trigonometric Fourier series and Laplace transformation over time are used. Laplace transforms of Green's functions are represented through the rational functions of Laplace transformation parameter. Transition into the space of the originals is carried out analytically using deductions and tables of operational calculus. Analytical expressions for Green functions of the problem under consideration are obtained. On the example of a simple supported three-component beam made of an alloy of zinc, copper, and aluminum, which is under the influence of mechanical load distributed along the length, the interaction of mechanical, temperature and diffusion fields is investigated. The influence of relaxation effects on the kinetics of heat and mass transfer is analyzed. The solution is presented in analytical form and in the form of graphs of the dependence of the desired fields of movement, temperature increments, and increments of concentration of medium components on time and coordinates. In conclusion, the main conclusions concerning influence of field connectivity and relaxation effects on the stress-strain state and heat and mass transfer in the bendable beam are given.

Keywords

thermoelastic diffusionLaplace transformFourier seriesGreen's functionsBernoulli – Euler beamunsteady problems.

НЕСТАЦИОНАРНЫЕ ТЕРМОУПРУГОДИФФУЗИОННЫЕ КОЛЕБАНИЯ БАЛКИ БЕРНУЛЛИ – ЭЙЛЕРА ПОД ДЕЙСТВИЕМ РАСПРЕДЕЛЕННОЙ ПОПЕРЕЧНОЙ НАГРУЗКИ

Рассматривается задача о нестационарных колебаниях балки Бернулли – Эйлера с учетом релаксации температурных и диффузионных процессов. Исходная математическая модель включает в себя систему уравнений нестационарных изгибных колебаний балки с учетом тепломассопереноса, которая получена из общей модели термомеханодиффузии для сплошных сред с помощью обобщенного принципа Даламбера. На основе полученных уравнений сформулирована постановка начально-краевой задачи об изгибе шарнирно-опертой ортотропной балки, находящейся под действием распределенных по поверхности термоупругодиффузионных возмущений. Решения задачи о нестационарных термоупругодиффузионных колебаниях балки ищется в интегральной форме. Ядрами интегральных представлений являются функции Грина, для нахождения которых используются разложения в тригонометрические ряды Фурье и преобразование Лапласа по времени. Трансформанты функций Грина представлены через рациональные функции параметра преобразования Лапласа. Переход в пространство оригиналов осуществляется аналитически с помощью вычетов и таблиц операционного исчисления. Получены аналитические выражения для функций Грина рассматриваемой задачи. На примере шарнирно-опертой трехкомпонентной балки, выполненной из сплава цинка, меди и алюминия, находящейся под действием распределенной по длине механической нагрузки, исследовано взаимодействие механического, температурного и диффузионного полей. Проанализировано влияние релаксационных эффектов на кинетику тепломассопереноса. Решение представлено в аналитической форме и в виде графиков зависимости искомых полей перемещения, приращений температуры и приращений концентрации компонент среды от времени и координат. В заключение приведены основные выводы о влиянии связанности полей и релаксационных эффектов на напряженно-деформированное состояние и тепломассоперенос в изгибаемой балке.

Ключевые слова

термоупругая диффузияпреобразование Лапласаряды Фурьефункции Гринабалка Бернулли – Эйлеранестационарные задачи.

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Article information

Published: 3 July 2023
Issue date: 3 July 2023
Issue: No. 3 (2023)