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On the combined asymptotic-tolerance modelling of dynamic problems for thin biperiodic cylindrical shells

Tomczyk B., Litawska A.

Vibrations in Physical Systems

2018

Vol. 29

art. no. 2018020

The objects of consideration are thin linearly elastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to formulate and discuss a new averaged general asymptotic-tolerance model for the analysis of selected dynamic problems for the shells under consideration. This model is derived by applying the combined modelling which includes two techniques: the asymptotic modelling procedure and a certain extended version of the known tolerance non-asymptotic modelling technique based on a new notion of weakly slowly-varying function. Contrary to the starting exact shell equations with highly oscillating, non-continuous and periodic coefficients, governing equations of the averaged combined model have constant coefficients depending also on a cell size. The differences between the general combined model proposed here and the corresponding known standard combined model derived by means of the more restrictive concept of slowly-varying functions are discussed.

Dynamic stability of micro-periodic cylindrical shells

Tomczyk B.

Mechanics and Mechanical Engineering

2010

Vol. 14, nr 2

351-374

The object of considerations are thin linear-elastic Kirchhoff-Love-type circular cylindrical shells having a micro-periodic structure along one direction tangent to the shell midsurface. Shells of this kind are called uniperiodic. The aim of this paper is twofold. First, we formulate an averaged non-asymptotic model for the analysis of dynamical stability of periodic shells under consideration, which has constant coefficients and takes into account the effect of a cell size on the overall shell behavior. This model is derived employing the tolerance modeling procedure. Second, we apply the obtained model to derivation of frequency equations being a starting point in the analysis of dynamical shell stability. The effect of the microstructure length on these frequency equations is discussed. The system of two the second-order ordinary differential frequency equations being a certain generalization of the known Mathieu equation is obtained. This system reduces to the Mathieu equation provided that the length-scale effect is neglected. Moreover, in the framework of the tolerance model proposed here the new additional higher-order free vibration frequencies and the new additional higher-order critical forces are derived. These frequencies and critical forces cannot be obtained from the asymptotic models commonly used for investigations of the shell stability.