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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

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.

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

Wierzbicki E.

Civil and Environmental Engineering Reports

2010

No. 5

95-99

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.

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.