Natural vibrations of thin conical panels of variable thickness

Abstract

A numerical analytical approach is proposed to study the natural vibrations of thin isotropic conical shells with varying thickness. The approach is based on the spline-approximation of unknown functions. Open shells (panels) with different boundary conditions are considered. The effect of variable thickness on the dynamic characteristics is analyzed Keywords: natural vibrations, thin isotropic conical shell, spline-collocation Introduction. Conical shells of variable thickness are widely used in many fields of modern engineering. An important aspect in making the shells strong is data on their natural vibrations. Recent trends in mathematics, mathematical physics, and mechanics are toward the wide use of spline-functions owing to their advantages over other methods of approximation. The main advantages of splines are their stability against local perturbations and the good convergence of the methods that employ splines as approximating functions. Using spline-functions in various variational, projective, and other discrete continuous methods makes it possible to refine the results of the classical polynomial approximation, to simplify considerably their numerical implementation, and to obtain a highly accurate solution.The present paper analyzes the natural vibrations of conical isotropic shells (panels) of variable thickness with different boundary conditions. Since the separation of variables is impossible for this class of shells, it is necessary to employ numerical methods.This paper proposes an efficient numerical method for studying the natural frequencies and modes of conical shells with varying thickness. 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. Problem Statement. Consider a problem on natural vibrations of an isotropic conical shell with varying thickness h x y ( , )in the curvilinear orthogonal coordinate system ( , ) s q , where s is the length of a meridian arc; q is the central angle in a parallel circle. In this case, the Lode parameters are: A = 1, B = r. The radii of principal curvatures R s and R q are: R s = 0, R r