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Drgania periodycznych struktur trójwarstwowych z rdzeniem Murakamiego

Marczak J., Jędrysiak J.

Modelowanie Inżynierskie

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2017

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T. 32, nr 63

97--106

PL

Przedmiotem rozważań jest analiza dynamiczna struktur trójwarstwowych o periodycznie zmiennych własnościach. Konstrukcja tego typu została wymodelowana jako układ dwóch płyt cienkich, połączonych ze sobą za pomocą materiału sprężystego Murakamiego. W tym celu rdzeń konstrukcji zastąpiono szeregiem jednorodnych prętów o stałych przekrojach, wykonujących niezależnie od siebie drgania podłużne. Dodatkowo założono, że każda z warstw konstrukcji może mieć periodycznie zmienną grubość i/lub własności materiałowe. Równania różniczkowe ruchu rozpatrywanej struktury są równaniami o periodycznie zmiennych, nieciągłych i silnie oscylujących współczynnikach, które z wykorzystaniem metody tolerancyjnego uśredniania da się sprowadzić do wygodnej do rozwiązania postaci o stałych współczynnikach.

EN

In this work the vibrations of periodic sandwich structures are considered. The model of such structure consists of two thin plates (outer layers), undergoing transverse vibrations, connected with each other by the inner layer, so called core, which is being modelled as a Murakami's type elastic material. Additionally, it is assumed, that every each of those layers can be characterised by certain periodic microstructure (both periodically varying thickness and material properties). The governing equations of such structure are partial differential equations with periodic, non-continuous and highly oscillating coefficients, which are very difficult to solve. In this work, the tolerance averaging technique is used to transform the mentioned system of equations into the form with constant coefficients, which is very convenient in solving many engineering problems.

2

Vibrations of periodic sandwich plates with inert core

Marczak J., Jędrysiak J.

Vibrations in Physical Systems

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2016

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Vol. 27

265--272

EN

In this note a free vibration analysis of periodic three-layered sandwich structures is performed. The equations of motion of such structures, which are derived basing on Kirchhoff's thin plate theory, contain periodic, non-continuous and highly oscillating coefficients, which makes them difficult to solve. In this work, the tolerance averaging technique is applied in order to transform the mentioned system of equations into a form with constant coefficients, which takes into account the effect of the microstructure size. The differences between two modelling procedures are shown and discussed. Eventually, formulas for free vibration frequencies of an exemplary 2D structure are derived and an analysis of influence of certain varying material properties is performed.

3

Analysis of non-stationary heat transfer in a hollow cylinder with functionally graded material properties performed by different research methods

Rabenda M. P.

Engineering Transactions

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2015

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Vol. 63, iss. 3

329--339

EN

A unidirectional non-stationary heat conduction in two-phase hollow cylinder is considered. The conductor is made of two-phase composites and has a smooth gradation of effective properties in the radial direction. Thus, we deal with a special case of functionally graded materials, FGM [1, 3, 4, 11, 12]. The mathematical model of the conductor is based on a tolerance averaging approach (TAA) [12]. Comparison of the results of tolerance model published in the article [7] with finite element method (FEM) calculated by Autodesk Simulation Multiphysics program is shown. The effect of geometry and material properties of conductor on the temperature field as well as different boundary conditions is examined.

4

Boundary effect behavior in the hexagonal-type biperiodic structures with rotational symmetry

Mazewska M., Kula D., Wierzbicki E.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 2

73--85

EN

The report presents a study of the thermal boundary effect behaviour in the hexagonal-type rigid conductors. Every hexagonal cell of the conductor is made of three rhombus parts. Some special cases of a considered conductor with rhombus parts built of two layers of different isotropic material are analyzed. The tolerance averaging approach is used as a tool of modeling. The most important result of the paper is specification of parameters which determine the character and the intensity of the boundary effect behavior. Introduced in the contribution the boundary effect intensity parameter is suggested to be treated as a certain measure of the intensity of boundary effect behavior.

5

On the tolerance averaging of heat conduction for periodic hexagonal-type composites

Wierzbicki E.

Civil and Environmental Engineering Reports

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2010

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No. 5

95-99

EN

The paper deals periodic composites which two-dimensional material structure is determined by the anisotropic conductivity matrix invariant with respect to the pair of translations such that the periodic cell determined by them coincide with the shapely hexagon. Moreover, it will be assumed that the material structure is invariant over the [pi]/3-rotations with the centers of every hexagon as the origin of the rotation. The main aim of this paper is to prove that the tolerance model of such composite, obtained by the orthogonalization method, is isotropic.

PL

W pracy zaproponowano równoważną postać tolerancyjnie uśrednionego modelu przewodnictwa ciepła kompozytu o mikroperiodycznej strukturze materialnej typu plastra miodu. Założono niezmienniczość tej mikrostruktury względem obrotów o kąt 2 [pi] 3 a także taką samą niezmienniczość stosowanych funkcji kształtu. W otrzymanej nowej postaci model ten ma współczynniki izotropowe.