Dynamics of flexible shells and Sharkovskiy′s periodicity

Vadim A Krysko,Natalya E Saveleva,Anton V Krysko,Jan Awrejcewicz

doi:10.1155/denm/2006/59709

Abstract

Complex vibration of flexible elastic shells subjected to transversal and sign‐changeable local load in the frame of nonlinear classical theory is studied. A transition from partial to ordinary differential equations is carried out using the higher‐order Bubnov‐Galerkin approach. Numerical analysis is performed applying theoretical background of nonlinear dynamics and qualitative theory of differential equations. Mainly the so‐called Sharkovskiy periodicity is studied.

Highlights

Complex vibration of flexible elastic shells subjected to transversal and sign-changeable local load in the frame of nonlinear classical theory is studied
In the shell body the middle surface is fixed for z = 0; axes ox and oy are directed along the main curvatures of this surface, whereas the oz axis is oriented into the curvature centre (Figure 1.1)
It is assumed that h(x, y) does not have first-order discontinuities, and the maximal thickness hmax ≡ h0 is sufficiently smaller than a smallest main curvature radius Rmin

Summary

Fundamental relations of flexible shells theory

We investigate a shallow shell which occupied three-dimensional subspace of a space R3 using a curvilinear system of coordinates x, y, z introduced in the following way. In the shell body the middle surface is fixed for z = 0; axes ox and oy are directed along the main curvatures of this surface, whereas the oz axis is oriented into the curvature centre (Figure 1.1). In the given coordinates the shell is treated as the three-dimensional object Ω defined as follows:. Normal coordinate to the middle surface is denoted by z. We assume that the quantity h0/Rmin can be neglected in comparison to 1. Shells satisfying this assumption are called thin shells. Note that an occurrence of nonhomogeneities of the boundary conditions occurred in second rows (for y = 0; ξ) exhibits existence of initial imperfections and stresses in the shell.

The Bubnov-Galerkin algorithm

Sharkovskiy periodicity exhibited by the obtained differential equations

Concluding remarks