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1
Content available The vibrations of non-symmetric periodic sandwich beams
Marczak Jakub
Vibrations in Physical Systems
|
2020
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Vol. 31, nr 2
art. no. 2020218
EN
Sandwich structures are certain specific type of composites, which are widely used in modern engineering. In this paper the vibration analysis of a specific type of sandwich beam is performed. The considered structure is not only non-symmetric towards its midplane, but also made of periodically varying isotropic materials. As a result, the governing equations of such complicated structure is characterised by periodic, non-continuous and highly oscillating coefficients. With the use of the tolerance averaging technique those equations are transformed into the form with constant coefficients. Eventually, a comparative simulations of free vibration analysis of several sandwich beams were conducted to verify the effectiveness and superiority of proposed calculation method over the FEM.
2
Content available Free vibrations spectrum of periodically inhomogeneous Rayleigh beams using the tolerance averaging technique
Świątek Marcin, Domagalski Łukasz, Jędrysiak Jarosław
Journal of Theoretical and Applied Mechanics
|
2019
|
Vol. 57 nr 1
141--154
EN
In this paper, linear-elastic Rayleigh beams with a periodic structure are considered. Dynamics of such beams is described by partial differential equations with non-continuous highly oscillating coefficients. The analysis of dynamic problems using the aforementioned equations is very often problematic to perform. Thus, other simplified models of Rayleigh beams are proposed. Some of these models are based on the concept of the effective stiffness. Among them, one can distinguish the theory of asymptotic homogenization. However, in these models, the size of the mesostructure parameter (the size of a periodicity cell) is often neglected. Therefore, a non-asymptotic averaged model of the periodic beam is introduced, called the tolerance model, which is derived by applying the tolerance averaging technique (TA). The obtained tolerance model equations have constant coefficients, and in contrast to other averaged models, some of them depend on the size of the periodicity cell.
3
Content available Effect of temperature on vibrations of laminated layer
Pazera E., Jędrysiak J.
Vibrations in Physical Systems
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2018
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Vol. 29
art. no. 2018032
EN
In this note the influence of temperature on vibrations of laminated layer made of two different materials is presented. The macroscopic properties of this layer are changing continuously along one direction x1, perpendicular to the laminas. To obtain the equations describing this problem, the tolerance averaging technique is used [1]. In this work, three models are proposed: the tolerance and the asymptotic-tolerance model, taking into account the effect of the microstructure size on the overall behaviour of this type of structures, and the asymptotic model, which equations omit this effect. To solve the equations of these three models the finite difference method is used.
4
Porównanie rozwiązań zagadnienia przewodnictwa cieplnego w przewodnikach warstwowych w modelach tolerancyjnym i asymptotycznym
Bagdasaryan V., Chalecki M.
Modelowanie Inżynierskie
|
2017
|
T. 34, nr 65
5--10
PL
Rozpatrywanie zagadnień początkowo-brzegowych dla przewodników warstwowych w ramach klasycznej teorii przewodnictwa cieplnego jest skomplikowane, ponieważ zagadnienia te są opisane przez równania różniczkowe o zmiennych i silnie oscylujących współczynnikach. W związku z tym poszukuje się modeli uproszczonych. W pracy porównano rozwiązania otrzymane w ramach dwóch uśrednionych modeli przewodnictwa cieplnego w periodycznych kompozytach warstwowych: modelu tolerancyjnego i jego wariantu asymptotycznego. Rozwiązania te otrzymano metodą różnic skończonych. Stwierdzono, że korzystanie z rozwiązań modelu asymptotycznego nie zawsze jest możliwe ze względu na zbyt dużą różnicę w wynikach względem modelu tolerancyjnego.
EN
Analysis of initial-boundary conditions for multilayered conductors within the framework of the classical theory of heat conduction is complex because these problems are described by differential equations with variable and highly oscillating coefficients. Due to this, simplified models are being sought. In this paper, the solutions obtained within the framework of two averaged models of heat conduction in multilayered materials have been compared: the tolerance model and its asymptotic version. These solutions have been obtained with use of the finite differences method. It has been stated that application of the asymptotic model is not always possible due to too large difference in results if compared to the tolerance model.
5
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.
6
Content available remote Tolerance modeling of dynamic behavior of thin plates with dense system of ribs in two directions
Michalak B., Rabenda M.
Archives of Civil and Mechanical Engineering
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2017
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Vol. 17, no. 2
222--230
EN
The subject of this paper are thin composite rectangular plate. The plates are made of two families of ribs and homogeneous material of a matrix. The main feature of the considered plates is, that a distance between the ribs is comparable to the thickness of the plate. The widths of the ribs can vary slowly in the midplane of the plate. This allows you to get a desirable frequency of natural vibrations of the plate. The formulation of averaged model equations is based on the tolerance averaging approach (Wozniak et al. 2008, 2010). The general results of the contribution are illustrated using the analysis of natural vibrations of the plates under consideration. It will be carried out validation of the obtained mathematical model by comparison of results from obtained model equations with results from finite elements method (Abaqus program).
7
Content available The hybrid method of finding the natural frequency spectrum for quasi-fractal plate bands
Wirowski A.
Journal of Applied Mathematics and Computational Mechanics
|
2015
|
Vol. 14, nr 4
153--165
EN
The work deals with the issues of modelling of multiscale composites. A group of composite structures with a quasi-fractal geometry has been defined. In order to model their dynamic properties, a simple engineering method has been proposed. It takes into account not only the parameters and proportions of the components of the composite, but also their mutual arrangement. The proposed method is demonstrated on several examples and its accuracy has been compared with the finite element method and the homogenization method.
8
Content available remote Vibrations of plate strips with internal periodic structure
Marczak J., Jędrysiak J.
Engineering Transactions
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2015
|
Vol. 63, iss. 1
93--108
EN
In this note vibrations of thin periodic plate strips with periodically distributed systems of three concentrated masses are analysed. Results of the non-asymptotic tolerance model are compared to those by the exact discrete model. In an example, these models are used to calculate lower and higher frequencies of the travelling wave related to the internal periodic structure.
9
Content available Analysis of vibrations of plate strip with concentrated masses using tolerance averaging technique
Marczak J., Jędrysiak J.
Vibrations in Physical Systems
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2014
|
Vol. 26
161--168
EN
In this note vibrations of thin periodic plate strips with periodically distributed system of two concentrated masses are analysed. Moreover, it is assumed that every concentrated mass is connected to a string, which cause the effect of damping in vibrations. Governing equation for such structure is defined as a differential equation with highly oscillating, periodic and non-continuous coefficients. In order to solve the equation, tolerance averaging technique is applied. As a result, governing equations with constant coefficients are obtained. In an example, derived model is used to calculate lower and higher frequencies of the travelling wave related to the internal periodic structure.
10
Dynamic modelling of thin plates made of longitudinally functionally graded material
Wirowski A.
Zeszyty Naukowe. Budownictwo / Politechnika Łódzka
|
2012
|
Z. 64
69-78
EN
Subject of this paper are thin plates with characteristic geometry: periodic in one direction and smoothly varying along another. The aim of the contribution is to formulate and investigate an averaged model describing the vibrations of this plate. Modelling procedure is based on the tolerance averaging technique. We are to analyze the plate in rectangular and polar coordinate systems.
PL
Przedmiotem pracy są płyty cienkie o określonej geometrii: periodyczne w jednym kierunku i wolnozmienne w drugim. Celem opracowania jest sformułowanie uśrednionego modelu opisującego drgania takiej płyty. Procedura modelowania jest oparta na technice tolerancyjnego uśrednienia. Analizowane płyty są w biegunowym i kartezjańskim układzie współrzędnych.
11
Content available Effect of composite ingredients on vibration frequencies of plates made of functionally graded material
Wirowski A.
Civil and Environmental Engineering Reports
|
2010
|
No. 5
165-177
EN
The subject of this paper are thin plates with characteristic material structure: periodic in selected direction and smoothly varying along another. The aim of the contribution is to formulate and apply averaged model describing the free vibrations of these plates. Modelling procedure is based on the tolerance averaging technique (TAT). We analyze the plate in the rectangular as well as in the cylindrical coordinate systems, respectively. We are to obtain numerical solutions of this problem, using finite difference method, and to analyze the interrelation between the ingredients distribution and the first frequency of free vibrations of these plates. The presented general results are illustrated by the analysis of natural frequencies for two cases of plates: a plate band and an annular plate.
PL
Przedmiotem niniejszej pracy są płyty cienkie posiadające charakterystyczną geometrię: periodyczną w jednym kierunku i zmieniającą się w sposób płynny w drugim. Celem rozważań jest zbudowanie modelu uśrednionego opisującego dynamiczne zachowanie tego typu płyty. Procedura modelowania jest oparta na technice tolerancyjnego uśredniania zaprezentowanej w pracy Woźniaka i Wierzbickiego [10]. Wyprowadzone równania modelu płyty są zapisane w układzie biegunowym dla płyty pierścieniowej oraz w układzie kartezjańskim dla przypadku pasma płytowego. Następnie zostało zaprezentowane rozwiązanie numeryczne za pomocą metody różnic skończonych oraz przeanalizowano wpływ udziału składników kompozytu na pierwszą częstość drgań własnych płyty.
12
Content available remote Parametric Vibrations of Thin Periodic Plates
Jędrysiak J.
Vibrations in Physical Systems
|
2004
|
Vol. 21
179--182
13
Content available remote On a Certain Model of Prestressed Medium Thickness Uniperiodic Plate
Baron E.
Vibrations in Physical Systems
|
2004
|
Vol. 21
101--104
14
Content available remote Boundary Efects in Dynamics of Laminated Elastic Half-space
Augustowska-Łacińska L.
Vibrations in Physical Systems
|
2004
|
Vol. 21
59--62
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