Solution describing the natural vibrations of rectangular shallow shells with varying thickness

Abstract

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.

Natural vibrations of thin conical panels of variable thickness

“…The following boundary conditions (6)-(9) are examined: G 1 : clamping at all edges (conditions (6) and (8)); G 2 : clamping at three edges (conditions (6) and (8) for q = b) and hinging at the fourth edge (condition (7) for q = 0); G 3 : clamping of the boundary s = const (conditions (6)) and hinging of the boundary q = const (conditions (9)). The relative differences among the frequencies obtained by the spline-collocation method with different number of collocation points (N = 10, 12, 14, 16) do not exceed 3%.…”

“…The method involves 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 stable numerical discrete orthogonalization in combination with step-by-step search [3,4]. Such an approach was used in [6][7][8][9].This method allows analyzing the natural vibrations of conical shells (panels) with arbitrarily varying thickness and complex boundary conditions. 1.…”

Identification of the parameters of the stress–strain problem for a multicomponent elastic body with an inclusion

“…approximation u n of the solution u Î U of problem (34), (7), the conjugate boundary-value problem has the form(14),…”

“…), we solve problem (40), (7) using the iterative process(9). The conjugate problem has the form(14). For an admissible increment Du of the element u Î U, the increment q = Dy of the solution y(u) can be found, based on (40), from the solution of the problem a w l u ( , ) ( q q = D , w), " w V To identify body forces, we can apply a parametric method,…”

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].…”