**Journal of Theoretical**

and Applied Mechanics

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

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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