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1
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.
2
Content available Nonlinear vibrations of a slender beam interacting with a periodic viscoelastic subsoil
Świątek M., Jędrysiak J., Domagalski Ł.
Vibrations in Physical Systems
|
2018
|
Vol. 29
art. no. 2018030
EN
The paper describes nonlinear vibrations of Euler-Bernoulli beams interacting with a periodic viscoelastic foundation. The original model equations with highly oscillating periodic coefficients are transformed using the tolerance modelling technique. Newly delivered equations have constant coefficients and describe macro-dynamics of the beam including the effect of the microstructure size. The main purpose of this paper is to propose an equivalent approximate model describing the nonlinear vibrations of a beam interacting with a periodic viscoelastic subsoil.
3
Content available Linear vibrations of periodic Timoshenko and Rayleigh beams
Świątek M., Jędrysiak J., Domagalski Ł.
Vibrations in Physical Systems
|
2018
|
Vol. 29
art. no. 2018035
EN
Elastic periodic structures with variable material and geometrical properties exhibit dynamic characteristics that are investigated in this contribution. The paper is devoted to analysis of geometrically linear vibrations of Rayleigh and Timoshenko beams with cross-sections and material properties periodically varying along the longitudinal axis. The period of inhomogeneity is assumed to be sufficiently small when compared to the beam length. Equations of motion in both beam theories under consideration have highly-oscillating coefficients. In order to derive the averaged model equations with constant coefficients for vibrations, the tolerance averaging approach is applied. The method of averaging differential operators with rapidly varying coefficients is applied to obtain averaged governing equations with constant coefficients. An assumed tolerance and indiscernibility relations and the definition of slowly varying function found the applied technique. Numerical results from the tolerance Rayleigh and Timoshenko beam model equations are compared.
4
Content available remote Free and forced large amplitude vibrations of periodically inhomogeneous slender beams
Domagalski Ł.
Archives of Civil and Mechanical Engineering
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2018
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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.
5
Content available Influence of substructure properties on natural vibrations of periodic Euler-Bernoulli beams
Świątek M., Jędrysiak J., Domagalski Ł.
Vibrations in Physical Systems
|
2016
|
Vol. 27
377--384
EN
In this paper there are considered vibrations of Euler-Bernoulli beams with geometrical and material properties periodically varying along the axis. The basic exact equations with highly oscillating periodic coefficients are replaced by the system of averaged equations with constant coefficients. The new model is based on the tolerance modelling technique, which describes macro-dynamics of the beam including the effect of the microstructure size. The purpose of this paper is to present an approximately equivalent model, which describe vibrations of periodic beams taking into account length of the periodicity cell.
6
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.
7
Content available A study on natural frequencies of Timoshenko beam with rapidly varying stiffness
Świątek M., Domagalski Ł., Jędrysiak J.
Vibrations in Physical Systems
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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.
8
Content available Nonlinear vibrations of periodic beams
Domagalski Ł., Jędrysiak J.
Journal of Theoretical and Applied Mechanics
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2016
|
Vol. 54 nr 4
1095--1108
EN
Geometrically nonlinear vibrations of beams with properties periodically varying along the axis are investigated. The tolerance method of averaging differential operators with highly oscillating coefficients is applied to obtain governing equations with constant coefficients. The proposed model describes dynamics of the beam with the effect of microstructure size. In an example, an analysis of undamped forced nonlinear vibrations of the periodic beam is shown. Moreover, the results obtained for undamped free vibrations of periodic beams by the tolerance model are justified by those results from the finite element method. These results can be used as a benchmark in similar problems.
9
Content available Nonlinear vibrations of periodic beams
Domagalski Ł., Jędrysiak J.
Vibrations in Physical Systems
|
2014
|
Vol. 26
73--78
EN
Geometrically nonlinear vibrations of beams with properties periodically varying along the axis are investigated. The tolerance method of averaging differential operators with highly oscillating coefficients is applied to obtain the governing equations with constant coefficients. The proposed model describes the dynamics of the beam with the effect of the microstructure size.
10
Geometrycznie nieliniowe zagadnienia stateczności cienkich płyt periodycznych
Domagalski Ł., Jędrysiak J.
Zeszyty Naukowe. Mechanika / Politechnika Opolska
|
2013
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z. 101
25--26
PL
W pracy rozpatrywane są geometrycznie nieliniowe zagadnienia stateczności cienkich prostokątnych płyt o budowie biperiodycznej. Pod pojęciem tym rozumie się tutaj struktury wykonane z jednorodnych izotropowych materiałów, gęsto użebrowane w obu kierunkach równoległych do krawędzi płyty, przy czym żebra usztywniające mogą być ukształtowane z materiału innego niż materiał płyty.
11
On a certain nonlinear model of thin periodic plates
Domagalski Ł., Gajdzicki M., Jędrysiak J.
Zeszyty Naukowe. Budownictwo / Politechnika Łódzka
|
2012
|
Z. 64
49-59
EN
The object under consideration are thin plates, which structure is periodic in planes parallel to the midplane. Plates of this kind consist of many small, repetitive elements, called periodicity cells, that can be treated as thin plates. The microstructure size is characterized by the diameter of the cell, which is called the microstructure parameter l. It is assumed that mechanical properties (bending and membrane stiffness tensors' components) of such plates are periodic, highly-oscillating, non-continuous functions. The main aim is to propose a mathematical model describing moderately large static deflections problem of considered plates, which is based on the tolerance modelling technique. A calculational example for a specific problem is included. The results are compared with results obtained within the linear model and with Finite Element Method.
PL
Rozważane są cienkie płyty o strukturze periodycznej w płaszczyznach równoległych do płaszczyzny środkowej. Płyty tego rodzaju składają się z wielu małych, powtarzalnych elementów, zwanych komórkami periodyczności, z których każda może być traktowana jak cienka płyta. Wielkość mikrostruktury jest charakteryzowana poprzez średnice (największy liniowy wymiar) komórki. Wymiar ten jest nazywany parametrem mikrostruktury i oznaczany przez /. Przyjęto, że własności mechaniczne płyty, reprezentowane przez składowe tensorów sztywności płytowych i tarczowych, są periodycznymi, nieciągłymi, silnie oscylującymi funkcjami. Głównym celem opracowania jest zaproponowanie matematycznego modelu opisującego zagadnienie umiarkowanie dużych ugięć rozważanych płyt, opartego na tzw. technice modelowania tolerancyjnego. Praca zawiera przykład obliczeniowy dla pewnego przypadku szczególnego. Dokonano porównania wyników uzyskanych w ramach proponowanego modelu nieliniowego, modelu liniowego oraz Metody Elementów Skończonych.
12
Content available Modelling of thin periodic plates subjected to large deflections
Domagalski Ł., Jędrysiak J.
Civil and Environmental Engineering Reports
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2010
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No. 5
133-148
EN
The objects under consideration are thin linear-elastic plates with periodic structure in planes parallel to the plate midplane, subjected to large (of the order of plate thickness) deflections. The main aim is to propose a mathematical model describing geometrically nonlinear problems of such plates, which is based on the tolerance averaging technique, cf. Woźniak et al. [3]. Results calculated for a special static problem by the tolerance model are compared with results obtained within the known tolerance linear model of thin plates.
PL
W pracy rozpatrywane są cienkie, liniowo-sprężyste płyty o budowie periodycznej w płaszczyznach równoległych do płaszczyzny środkowej. Zagadnienia statyki i dynamiki tego rodzaju płyt w zakresie dużych ugięć opisane są układem równań różniczkowych nieliniowych o silnie oscylujących, periodycznych, nieciągłych współczynnikach (por. książka pod red. Woźniaka i in. [3]). W celu otrzymania równań o stałych współczynnikach zastosowano tu technikę tolerancyjnego uśredniania, omówioną w książce pod red. Woźniaka, Michalaka i Jędrysiaka [5]. Zaproponowano nieliniowy model tolerancyjny, opisujący nieliniowo-geometryczne zagadnienia cienkich płyt periodycznych. Model ten zastosowano do wyznaczenia ugięć dla danego obciążenia, a otrzymane wyniki porównano z wynikami uzyskanymi w ramach liniowego modelu tolerancyjnego.
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