Journal of Theoretical
and Applied Mechanics

57, 1, pp. 141-154, Warsaw 2019
DOI: 10.15632/jtam-pl.57.1.141

### Free vibrations spectrum of periodically ingomogeneous Rayleigh beams using the tolerance averaging technique

Marcin Świątek, Łukasz Domagalski, Jarosław Jędrysiak
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.
Keywords: periodicity cell, Rayleigh beam, tolerance averaging technique

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