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1
Content available Vibrations of microstructured beams with axial force
Jędrysiak Jarosław
Vibrations in Physical Systems
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2020
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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.
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
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2018
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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
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2018
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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 Influence of substructure properties on natural vibrations of periodic Euler-Bernoulli beams
Świątek M., Jędrysiak J., Domagalski Ł.
Vibrations in Physical Systems
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2016
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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.
5
Content available An analytical-numerical approach to analysis of large amplitude vibrations of slender periodic beams
Domagalski Ł.
Vibrations in Physical Systems
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2016
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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.
6
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.
7
Content available Nonlinear vibrations of periodic beams
Domagalski Ł., Jędrysiak J.
Vibrations in Physical Systems
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2014
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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.
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