Dynamic Stability of Micro{Periodic Cylindrical Shells
Barbara Tomczyk · Mechanics and Mechanical Engineering · 2010
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.