The natural vibrations of a corrugated elastic orthotropic cylindrical shell with a directrix perpendicular to its edges that are free are examined Keywords: corrugated elastic orthotropic cylindrical shell, natural vibrations, momentless stress state, absence of bending stiffness Introduction. Many important research and technology problems involve study of the vibrations of and wave propagation in deformed solid media. These are in particular problems of seismic prospecting, aircraft construction, shipbuilding, instrument making, dynamics of power-engineering structures, etc. Rayleigh's work [28] was the first to study elastic surface waves. He discovered elastic waves propagating along the free boundary of a half-space and having amplitude quickly decreasing with depth. Such waves observed in elastic bodies of various geometries are commonly referred to as Rayleigh waves. Waves localized near the free boundary of a semi-infinite plate and waves propagating along a semi-infinite cylindrical shell and decaying with distance from the free end are of Rayleigh type too [7]. In a semi-infinite plate, according to a Kirchhoff-Love hypothesis, there exist independent planar and bending waves localized along the free boundary [2,3,6,21,22]. If the plate is curved, these two waves appear coupled, giving rise to new two (preferential bending and preferential tangential) oscillations localized at the edge [13][14][15]. Of special interest are problems on cylindrical shells of varying curvature. Such problems are solved using various analytic and numerical methods [1,9,10,24,[29][30][31][32]. The natural vibrations of a semi-infinite momentless cylindrical shell damped with distance from its free edge are examined in [5,12,16,17]. The natural vibrations of a cantilever corrugated orthotropic momentless cylindrical shell are studied in [18].The present paper is concerned with the natural vibrations of a corrugated (and closed) orthotropic momentless cylindrical shell with free edges. It is assumed that the generatrices are perpendicular to the edges and the squared curvature of the directrix can be expanded into a series: R k r r km km m m m -= ¥ = + + ae è ç ç ö ø ÷ ÷ å 2 2 0 1 2 / cos sin b r b ,where b is the directed arc length of the directrix. Note that the class of cylindrical shells under consideration includes closed cylindrical shells with free edges and arbitrary smooth directrices. In this case, k n s = 2 0 p / , where s is the total length of the directrix, n N 0 Î . We have derived dispersion equations and established asymptotic relationships between these equations and the dispersion equations for a shell structure consisting of countably identical open orthotropic circular cylindrical shells and an orthotropic strip plate and the dispersion equations for a corrugated semi-infinite orthotropic momentless cylindrical shell with free edge. We will present approximate values of the dimensionless natural frequencies and damping factors for cylindrical shells having lengths l = 5 and l = 15, directrices y a cx b c...
The forced vibrations of a cylindrical orthotropic shell are studied. Two types of boundary conditions on the outer surface are examined considering that the displacement vector prescribed on the inner surface varies harmonically with time. Asymptotic solutions of associated dynamic equations of three-dimensional elasticity are found. Amplitudes of forced vibrations are determined and conditions under which resonance occurs are established. Boundary-layer functions are defined. The rate of their decrease with distance from the ends inside the shell is determined. A procedure of joining solutions for the internal boundary-layer problem is outlined in the case for the, if clamping boundary conditions are prescribed at the ends Keywords: orthotropic shell, forced vibrations, asymptotic solution, boundary layer Introduction. The classical theory of plates and shells and the available refined theories consider only one class of problems (first boundary-value problem of elasticity) where components of the stress tensor are specified on the faces of plates and shells. There are many applied problems where other conditions (e.g., displacement vector) or mixed conditions (nonclassical boundary-value problems for thin bodies) are specified on the faces of a thin body (beam, plate, or shell). Such problems arise, for example, in foundation engineering (when analyzing the effect of earthquakes on structures), vibroseis exploration, runway design, joint of compliant thin-walled members to more rigid ones, etc. A direct verification shows that the hypotheses of the classical theory of beams, plates, and shells and the modern refined theories cannot be used to solve problems of these classes because the equations of these theories fail to satisfy the above boundary conditions. Such problems were mainly solved with methods of the elasticity theory for infinite strips and layers and the methods of integral transformations [16,23] and potential [20]. The numerical analytical method is efficient. However, it is natural to apply the asymptotic method to beams, plates, and shells because one of their dimensions is radically different from the other two, which gives rise to a small geometrical parameter when dimensionless coordinates are used. The transformed equations are singularly perturbed. Their solution consists of the solutions of the internal problem and boundary-layer problem [15,21,22]. Asymptotic theories of isotropic plates and shells [17] and asymptotic theories of anisotropic beams, plates, and shells [1] corresponding to the classical theory (i.e., static conditions of the first boundary-value problem of elasticity prescribed on the faces) have been set up. The results obtained were compared with those produced by the classical and refined theories. Mathematically exact solutions for some classes of problems, including the problem for a strip were obtained. This made it possible to prove that the Saint-Venant principle is valid for the first boundary-value problem [1,13].The asymptotic method turned out to be efficient ...
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