Vibrations of thin cylindrical shells with a periodic structure

Abstract

Free vibrations of thin linear-elastic Kirchhoff-Love cylindrical shells, having a periodic structure along one direction tangent to the shell midsurface, is considered. In order to take into account the effect of the periodicity cell size in this problem, a new averaged non-asymptotic model of such shells, proposed by Tomczyk (2006), is applied. The new additional higher-order free vibration frequencies dependent on the microstructure size will be derived and discussed. PreliminariesThe new non-asymptotic mathematical model for problems of dynamics and dynamical stability of thin linear-elastic KirchhoffLove-type cylindrical shells with periodically variable thickness as well as periodically variable inertial and elastic material properties along one direction tangent to the shell midsurface has been proposed in [2]. In order to derive this model the tolerance averaging technique, presented in [3], was applied. Shells under consideration are called uniperiodic.The aim of this contribution is to apply the model equations derived in [2] to investigate the effect of a cell size on free vibrations of a circular periodically stiffened shell as shown in Fig.1. The length-scale effect is neglected in commonly used homogenized models of such shells, derived by means of asymptotic methods, [1].Definestand for a Cartesian orthogonal coordinate system in the physical space E 3 . A cylindrical shell midsurface M will be given by its parametric representation M ≡ {x ∈ E 3 : x = r(ξ), (ξ) ∈ Ω}, where r(·) is the smooth function such that ∂ 1 r · ∂ 2 r = 0, ∂ 1 r · ∂ 1 r = ∂ 2 r · ∂ 2 r = 1. It means that on M we have introduced the orthonormal parametrization and hence L 1 , L 2 are length dimensions of M . Let the shell, considered here, be closed and circular. It means that now L 1 = 2πρ, where ρ is the midsurface curvature radius. Let δ stand for the constant shell thickness. The shell is reinforced by two families of ribs, which are periodically and densely distributed in circumferential direction, cf. Fig. 1. The stiffeners of both kinds are assumed to have constant cross-sections with A 1 , A 2 as their areas and with I 1 , I 2 as their moments of inertia. Let a 1 , a 2 be the widths of the ribs. It is assumed that both the shell and stiffeners are made of homogeneous isotropic materials and let us denote by E, ν Young's modulus and Poisson's ratio of the shell material, respectively, and by E 1 , E 2 Young's moduli of the rib materials. Let µ 0 and µ 1 , µ 2 stand for the shell mass density and for the mass densities of the stiffeners, respectively, cf. Fig. 1. On 0ξ1 ξ 2 -plane we define λ as the period of the stiffened shell structure in ξ 1 -direction, cf. Fig. 1. The period λ satisfies the conditions: λ/δ max 1, λ/ρ min 1 and λ/L 1 1. We also assume that L 2 > L 1 and hence λ/L 2 1. The basic cell ≡ [−λ/2, λ/2] × {0} has a symmetry axis for η 1 = 0, where η 1 ∈ [−λ/2, λ/2]. It is assumed that the shell is simply supported on the edges ξ 2 = 0, ξ 2 = L 2 . The tensile and bending rigidities of the stiffeners are ...