Natural vibrations of shallow cylindrical shells with rectangular plan and varying thickness are studied using a spline-approximation method developed previously. Computation is carried out for different types of boundary conditions. The effect of the curvature of the midsurface on the natural frequencies is examined. The natural frequencies of shells with constant and varying thickness are compared Introduction. Shallow shells of various shapes are widely used as structural members in modern engineering and building structures. The operating conditions for these structures impose certain requirements on their strength and reliability. In this connection, efficient numerical and experimental methods for the determination of the load-bearing capacity and, in particular, resonant frequencies of such structures take on special significance.Of interest are the natural vibrations of rectangular (in plan) shallow shells with varying thickness and different boundary conditions. For shells of constant thickness with hinged edges, it is possible to find a closed-form solution [5,6]. If the edges are clamped, then the variables in the original equations of motion cannot be separated and, therefore, numerical methods should be applied. There are just a few publications devoted to this class of problems [4,[11][12][13]. This is because their solution involves computational difficulties.Spline functions have recently been used to study the mechanical behavior of plates and shells. Their main advantages are: -stability against local perturbations; i.e., the local behavior of splines in the neighborhood of a point does not influence their overall behavior, in contrast to, for example, polynomial approximation; -better convergence than that of polynomial approximation; -simple and convenient computer implementation. This paper proposes an efficient numerical technique for studying the natural frequencies and modes of shallow rectangular (in plan) shells of varying thickness. The technique is based on spline-approximation in one coordinate direction and solution of a boundary-value 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. The shell material is generally anisotropic.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 [1,[7][8][9][10].With such an approach, we can study the natural vibrations of a wide class of isotropic and anisotropic shallow shells with arbitrarily varying thickness and complex boundary conditions. The objective of the present paper is to study the natural vibrations of elastic rectangular (in plan) shallow shells with varying thickness on the basis of spline-approximation.
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].
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