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1
Content available Free vibrations of microstructured functionally graded plate band with clamped edges
Kaźmierczak-Sobińska Magda, Jędrysiak Jarosław
Vibrations in Physical Systems
|
2021
|
Vol. 32, nr 2
art. no. 2021216
EN
In this paper there are presented free vibrations of thin functionally graded plate band. This kind of plates has tolerance-periodic microstructure on the microlevel in planes parallel to the plate midplane. Dynamic problems of plates of this kind are described by partial differential equations with highly oscillating, tolerance-periodic, non-continuous coefficients. Thus, there are proposed here two models describing these plates by equations with smooth, slowly-varying coefficients. As an example there are analyses of free vibration frequencies for thin functionally graded plate band clamped on both edges. Using the known Ritz method the frequencies are obtained in the framework of proposed two models - the tolerance model and the asymptotic model.
2
Content available Vibrations of microstructured beams with axial force
Jędrysiak Jarosław
Vibrations in Physical Systems
|
2020
|
Vol. 31, nr 2
art. no. 2020208
EN
In this contribution there are considered vibrations of microstructured periodic slender beams, with axial force. In order to analyse the effect of the microstructure size of the beams on their vibrations the tolerance modelling method is applied. Using this method there are derived governing equations of two tolerance models - general and standard, base on two various concepts - weakly-slowly-varying functions and slowly-varying functions. These models are applied to obtain formulas of lower order and higher order frequencies with influence of the axial force. To evaluate these results of the modelling the formula of lower order frequencies in the framework of the asymptotic model (neglecting the effect of the microstructure size) is also derived.
3
Content available Effect of fluctuation shape functions on vibrations of laminated structures
Kubacka Ewelina
Vibrations in Physical Systems
|
2020
|
Vol. 31, nr 2
art. no. 2020213
EN
In this note, the influence of the fluctuation shape functions on vibrations of the periodic laminate is analysed. The structure, used to show this impact, is the composite, consisting of the layers made of components differ in material properties like a specific heat or a thermal conductivity. The periodic laminate is microscopically heterogenous and to analyse this laminate, the tolerance averaging technique is used, therefore the influence of the thickness of the layer can be allow. One of the concepts introduced by tolerance modelling, is the fluctuation shape function, affecting on the results. The fluctuation shape function is assumed a priori and the character of vibrations is dependent on this function.
4
Content available Tolerance modelling of nonstationary problems of microheterogeneous media and structures
Jędrysiak J., Domagalski Ł., Marczak J., Pazera E.
Vibrations in Physical Systems
|
2018
|
Vol. 29
art. no. 2018001
EN
In this note a certain review of applications of a non-asymptotic modelling approach, called the tolerance modelling, is presented. Some objects and thermomechanical problems are shown, with a general outline of this method and an example of application for nonlinear vibrations of periodic beams.
5
Content available On the combined asymptotic-tolerance modelling of dynamic problems for thin biperiodic cylindrical shells
Tomczyk B., Litawska A.
Vibrations in Physical Systems
|
2018
|
Vol. 29
art. no. 2018020
EN
The objects of consideration are thin linearly elastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to formulate and discuss a new averaged general asymptotic-tolerance model for the analysis of selected dynamic problems for the shells under consideration. This model is derived by applying the combined modelling which includes two techniques: the asymptotic modelling procedure and a certain extended version of the known tolerance non-asymptotic modelling technique based on a new notion of weakly slowly-varying function. Contrary to the starting exact shell equations with highly oscillating, non-continuous and periodic coefficients, governing equations of the averaged combined model have constant coefficients depending also on a cell size. The differences between the general combined model proposed here and the corresponding known standard combined model derived by means of the more restrictive concept of slowly-varying functions are discussed.
6
Content available Elastostatic problems in multicomponent, multilayered periodic composites
Wągrowska M., Szlachetka O., Bagdasaryan V.
Przegląd Naukowy Inżynieria i Kształtowanie Środowiska
|
2018
|
Vol. 27, No. 1
9--18
EN
The object of the analysis is a two-dimensional elastostatic problem for multicomponent, multilayered periodic composites. The equations of equilibrium for this composite are obtained within the framework of tolerance modelling procedure. The paper presents two examples of solutions of boundary value problems.
7
Content available remote On thermoelasticity in FGL - tolerance averaging technique
Pazera E., Ostrowski P., Jędrysiak J.
Mechanics and Mechanical Engineering
|
2018
|
Vol. 22, nr 3
703--717
EN
In this paper the problem of linear thermoelasticity in a laminate with functional gradation of properties is considered. In micro level this laminate is made of two different materials, microlaminas, distributed non-periodically but also not randomly along one of directions, what in macro level results in aforementioned functionally gradation of laminate properties. In order to describe behavior of such structure, equations of two models are here presented - the tolerance and the tolerance-asymptotic model. Both are obtained by the tolerance averaging technique. The basic aim of this work is to analyse the influence of some terms from these averaged equations on the distribution and the values of the displacements and the temperature functions. To solve the equations of two proposed models the finite difference method is used.
8
Content available remote Micro-vibrations and wave propagation in biperiodic cylindrical shells
Tomczyk B., Litawska A.
Mechanics and Mechanical Engineering
|
2018
|
Vol. 22, nr 3
789--807
EN
The objects of consideration are thin linearly elastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to study a certain long wave propagation problem related to micro-fluctuations of displacement field caused by a periodic structure of the shells. This micro-dynamic problem will be analysed in the framework of a certain mathematical averaged model derived by means of the combined modelling procedure. The combined modelling applied here includes two techniques: the asymptotic modelling procedure and a certain extended version of the known tolerance non-asymptotic modelling technique based on a new notion of weakly slowly-varying function. Both these procedures are conjugated with themselves under special conditions. Contrary to the starting exact shell equations with highly oscillating, non-continuous and periodic coefficients, governing equations of the averaged combined model have constant coefficients depending also on a cell size. It will be shown that the micro-periodic heterogeneity of the shells leads to exponential micro-vibrations and to exponential waves as well as to dispersion effects, which cannot be analysed in the framework of the asymptotic models commonly used for investigations of vibrations and wave propagation in the periodic structures.
9
Content available remote Fourier variant homogenization treatment of single impulse boundary effect behaviour
Kula D., Wierzbicki E., Witkowska-Dobrev J., Wodzyński Ł.
Mechanics and Mechanical Engineering
|
2018
|
Vol. 22, nr 3
683--690
EN
Boundary effect behavior understood as near-boundary suppression of boundary fluctuation loads is described in various ways depending on the mathematical representation of solutions and the type of the center. In the case of periodic composites, the homogenization method is decisive here. In the framework of the Tolerance Averaging Approach, developed by prof. Cz. Wo´zniak leading to an approximate model of phenomena related to periodic composites this effect is described by a homogeneous part of differential equation for fluctuation amplitudes and usually this approximate description of the boundary effect behavior is restricted to a single fluctuation. In this paper, contrary to the previous elaborations, the boundary effect is developed in the variant of the tolerance thermal conductivity model in which the temperature field is represented by the Fourier expansions composed by an average temperature with infinite number of Fourier terms imposed on the average temperature as tolerance fluctuation suppressed in the framework of the boundary effect.
10
Content available remote Fourier variant homogenization of the heat transfer processes in periodic composites
Wierzbicki E., Kula D., Wodzyński Ł.
Mechanics and Mechanical Engineering
|
2018
|
Vol. 22, nr 3
719--726
EN
The well-known parabolic Heat Transfer Equation is a simplest recognized description of phenomena related to the heat conductivity in solids with microstructure. However, it is a tool difficult to use due to the discontinuity of coefficients appearing here. The purpose of the paper is to reformulate this equation to the form that allows to represent solutions in the form of Fourier’s expansions. This equivalent re-formulation has the form of infinite number of equations with Fourier coefficients in expansion of the temperaturę field as the basic unknowns. The first term in Fourier representation, being an average temperature field, should satisfy the well-known parabolic heat conduction equation with Fourier coefficients as fields controlling average temperature behavior. The proposed description takes into account changes of the composite periodicity accompanying changes in the variable perpendicular to the surfaces separating components, concerning FGM-type materials and can be treated as the asymptotic version of Heat Transfer Equation obtained as a result of a certain limit passage where the cell size remains unchanged.
11
Content available remote Displacements caused by the temperature in multicomponent, multi-layered periodic material structures
Bagdasaryan V., Wągrowska M., Szlachetka O.
Mechanics and Mechanical Engineering
|
2018
|
Vol. 22, nr 3
809--819
EN
The present study aims to analyse a two-dimensional problem of displacements in theory of thermal stresses for multicomponent, multi-layered periodic composites. The model equations are obtained within the framework of the tolerance modelling procedure. These equations allow to determine the distribution of displacements caused by the temperature field in the theory of thermal stresses. The paper presents an example of a solution of a boundary value problem.
12
Content available remote Free and forced large amplitude vibrations of periodically inhomogeneous slender beams
Domagalski Ł.
Archives of Civil and Mechanical Engineering
|
2018
|
Vol. 18, no. 4
1506--1519
EN
Considered are free and forced transverse vibrations of slender periodic beams of finite length. It is assumed that the vibration amplitude is of the order of cross-section dimensions, still much smaller than the beam length. An averaged non-asymptotic model is proposed as a tool in analysis. The description is based on the tolerance approach to averaging of differential operators, using the concept of weakly slowly-varying function. The resulting differential equations with constant coefficients involve the effect of periodicity cell length. The model is verified by comparison of linear frequencies and mode shapes with Finite Element Method results, and then applied in analysis of free and forced vibrations of beam with variable cross-section. The method employed in obtaining the solution involves Galerkin orthogonalization and Runge–Kutta (RKF45) method. The results of nonlinear vibrations analysis are presented by backbone and amplitude-frequency response curves, time series, Poincare sections and bifurcation diagrams.
13
Niestacjonarne zjawisko termosprężystości w nieperiodycznym laminacie
Jędrysiak J., Pazera E
Modelowanie Inżynierskie
|
2017
|
T. 32, nr 63
65--71
PL
W niniejszej pracy rozpatrywany jest efekt termosprężystości w zagadnieniu niestacjonarnym w laminacie nieperiodycznym. Własności takiego laminatu na poziomie makro zmieniają się w sposób ciągły wzdłuż osi prostopadłej do lamin, natomiast na poziomie mikro są opisane funkcjami tolerancyjnie-periodycznymi, nieciągłymi. W celu otrzymania równań o ciągłych współczynnikach funkcyjnych zastosowano metodę tolerancyjnego modelowania, czyli tzw. tolerancyjne modelowanie, oraz modelowanie asymptotyczne. Pierwsze z tych podejść pozwala uwzględnić w wyprowadzonych równaniach wpływ wielkości mikrostruktury laminatu. Zastosowanie otrzymanych równań pokazano na przykładzie jednokierunkowego przepływu ciepła prostopadle do lamin.
EN
In this work the thermoelasticity effect in a non-stationary heat problem for non-periodic laminates is considered. On the macro-level properties of this laminate are continuously varied along a direction normal to laminas, but on the micro-level they are described by tolerance-periodic, non-continuous functions. In order to obtain equations with continuous coefficients the tolerance method is applied, the tolerance modelling and the asymptotic modelling. The first of these approaches allows to take into account the effect of the microstructure size of the laminate in derived equations. An application of these model equations is shown on an example of one-directional problem along a direction perpendicular to laminas.
14
Content available remote A new tolerance model of dynamic thermoelastic problems for thin uniperiodic cylindrical shells
Tomczyk B., Ślęzowski B.
Engineering Transactions
|
2017
|
Vol. 65, iss. 1
179--191
EN
The objects of consideration are thin linearly thermo-elastic Kirchhoff-Love-type circular cylindrical shells having a periodically micro-inhomogeneous structure in circumferential direction (uniperiodic shells). The aim of this note is to formulate and discuss a new non-asymptotic averaged model for the analysis of selected dynamic thermoelastic problems for these shells. Contrary to the starting exact shell equations with highly oscillating, non-continuous and periodic coefficients, the proposed tolerance model equations have constant coefficients depending also on a cell size. Hence, an important advantage of this model is that it makes it possible to investigate the effect of a period of inhomogeneity on the global shell thermodynamics (the length-scale effect). This effect is neglected in the known homogenized models derived by asymptotic methods.
15
Content available remote A new asymptotic-tolerance model of dynamic problems for transversally graded cylindrical shells
Tomczyk B., Szczerba P.
Engineering Transactions
|
2017
|
Vol. 65, iss. 1
171--178
EN
The objects of consideration are thin linearly elastic Kirchhoff-Love-type open circular cylindrical shells having a functionally graded macrostructure and a tolerance-periodic microstructure in circumferential direction. The aim of this note is to formulate and discuss a new non-asymptotic averaged model for the analysis of selected dynamic problems for these shells. The proposed asymptotic-tolerance model equations have continuous and slowly varying coefficients depending also on a cell size. An important advantage of this model is that it makes it possible to study micro-dynamics of tolerance-periodic shells independently of their macro-dynamics.
16
Content available Vibrations of non-periodic thermoelastic laminates
Jędrysiak J., Pazera E.
Vibrations in Physical Systems
|
2016
|
Vol. 27
175--180
EN
Vibrations of non-periodic thermoelastic laminates, which can be treated as made of functionally graded material with macroscopic properties changing continuously along direction, x1, perpendicular to the laminas on the macrolevel are considered. Three models of these laminates are presented: the tolerance, the asymptotic-tolerance and the asymptotic. Governing equations of two first of them involve terms dependent of the microstructure size. Hence, these models (the tolerance, the asymptotic-tolerance) describe the effect of the microstructure. Averaged governing equations of these laminates can be obtained using the tolerance modelling technique, cf. Jędrysiak [1]. Because the model equations have functional, but slowly-varying coefficients calculations for examples can be made numerically or by using approximated methods.
17
Content available Higher order vibrations of thin periodic plate bands with various boundary conditions
Jędrysiak J., Wyrwa P.
Vibrations in Physical Systems
|
2016
|
Vol. 27
181--186
EN
In this contribution there are considered thin periodic plates. The tolerance averaging method, cf. [12, 13, 4], is applied to model problems of vibrations of these plates. Hence, the effect of the microstructure size is taken into account in model equations of the tolerance model. Calculations are made for periodic plate bands using this model and the asymptotic model for various boundary conditions.
18
Content available Free vibrations of medium thickness microstructured plates
Jędrysiak J.
Vibrations in Physical Systems
|
2016
|
Vol. 27
169--174
EN
A problem of free vibrations of medium thickness microstructured plates, which can be treated as made of functionally graded material on the macrolevel is presented. The size of the microstructure of these plates is of an order of the plate thickness. Averaged governing equations of these plates can be obtained using the tolerance modelling technique, cf. [18, 19, 9]. Because, the derived tolerance model equations have the terms dependent of the microstructure size, this model describes the effect of the microstructure size. Results can be evaluated introducing the asymptotic model. Calculated results can be compared to those from the finite element method or a similar tolerance model of thin plates, cf. [9].
19
Content available An analytical-numerical approach to analysis of large amplitude vibrations of slender periodic beams
Domagalski Ł.
Vibrations in Physical Systems
|
2016
|
Vol. 27
99--106
EN
The paper is devoted to analysis of geometrically nonlinear vibrations of beams with geometric and material properties periodically varying along the axis. The 1-D Euler-Bernoulli theory of beams with von Kármán nonlinearity is applied. An analytical-numerical model based on non-asymptotic tolerance modelling approach and Galerkin method is applied. The linear natural frequencies and mode shapes are determined and the results are confirmed by comparison with a finite element model. Forced damped vibrations analysis in the large deflection range is performed to illustrate complex behaviour of the system.
20
Content available A study on natural frequencies of Timoshenko beam with rapidly varying stiffness
Świątek M., Domagalski Ł., Jędrysiak J.
Vibrations in Physical Systems
|
2016
|
Vol. 27
369--376
EN
Vibrations of Timoshenko beams with properties periodically varying along the axis are under consideration. The tolerance method of averaging differential operators with highly oscillating coefficients is applied to obtain the governing equations with constant coefficients. The dynamics of Timoshenko beam with the effect of the cell length is described. A asymptotic model is then constructed, which is further studied in analysis of the low order natural frequencies. The proposed model is able to describe dynamics of beams made of non-slender cells.
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