Free vibrations of rectangular orthotropic shallow shells with varying thickness

Abstract

The paper proposes a numerical-analytic approach to studying the free vibrations of orthotropic shallow shells with double curvature and rectangular planform. The approach is based on the spline-approximation of unknown functions. Calculations are carried out for different types of boundary conditions. The influence of the mid-surface curvature and variable thickness on the behavior of dynamic characteristics is studied Keywords: free vibrations, shallow shell, double curvature, spline-collocation Introduction. Shallow shells of various shapes are widely used as rational elements in many branches such as building, aircraft construction, shipbuilding, rocket and missile engineering, etc. Anisotropic shallow shells are of wide use in high-strength and reliable structures intended to operate under severe conditions. One of the important aspects of making such elastic bodies durable is obtaining information on their free vibrations.Recent trends have been toward the use of spline functions to solve problems in computational mathematics, mathematical physics, and mechanics. This is due to the advantages of splines over other approximations:-stability against local perturbations; i.e., the behavior of a spline near a point does not influence its overall behavior, in contrast to, for example, polynomial approximation; -better convergence than that of polynomial approximation; -simple and convenient computer implementation. When used in various variational, projective, and other methods, spline functions perform better than classical polynomials, substantially simplify numerical implementation, and produce solutions with high accuracy. This paper studies the free vibrations of orthotropic shallow shells with rectangular planform, double curvature, varying thickness, and boundary conditions of different types. A closed-form solution can be found for hinged shells of constant thickness [7,8]. If, however, shells are clamped, the variables of the original equations of motion cannot be separated, which necessitates using numerical methods. There are few publications on this class of problems [5,[13][14][15]. This is because of computational difficulties. The principles of anisotropic elasticity theory and anisotropic shell theory are outlined in the fundamental monographs [1,4,6].In what follows, we present an efficient numerical technique to study the natural frequencies and modes of orthotropic shallow shells with rectangular planform, double curvature, and varying thickness. The technique employs spline-approximation in one of the coordinate directions, followed by solution of an eigenvalue problem for systems of ordinary differential equations of high order with variable coefficients by the stable discrete-orthogonalization method in combination with step-by-step search.Noteworthy is the series of publications where the spline-approximation was used to analyze the stress-strain state of shells of different structure and the natural vibrations of plates [2,[9][10][11][12].

Large amplitude free vibration of orthotropic shallow shells of complex shapes with variable thickness

“…, for a clamped spherical panel versus results reported in reference [4] is given in Table 1. In what follows we study the influence of parameter variation .…”

“…In what follows we study the influence of parameter variation . This problem has been solved in [4], using a spline -approximation to the assumed solution. One may see that the difference between our results and those given in [4] is less than 1.5%.…”

“…In practice, these elements can have a variable thickness, different form of the middle surface and boundary conditions, as well as different orientation of the anisotropy axes. The studies of linear vibrations of anisotropic shells have attracted the attention of many researchers for a long time [4,6,7,9,10]. Great progress has been made over the past decades to develop numerical approximate methods as the most effective tools for studying nonlinear vibrations of the composite plates and shallow shells [1-3, 5, 7, 9-11].…”

Static and Dynamic Problems for Anisotropic Inhomogeneous Shells with Variable Parameters and Their Numerical Solution (Review)

“…Problems of the free vibrations of plates and shallow shells with variable thickness were solved using the proposed approach and the classical theory in [12,34,35,50].…”

Free vibrations of shallow nonthin shells with variable thickness and rectangular planform

“…by solving a boundary-value eigenvalue problem for systems of ordinary differential equations of high order with variable coefficients by stable numerical discrete orthogonalization in combination with incremental method [2,3,7,8].…”