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

Abstract

The free vibrations of shallow orthotropic shells with variable thickness and rectangular planform are studied. The shear strains are taken into account. The spline approximation of unknown functions is used. The natural frequencies are calculated for different boundary conditions. The dependence of the natural frequencies on the curvature of the midsurface is examined. The natural frequencies of shells with constant and variable thickness are compared Keywords: shallow orthotropic shell with rectangular planform, shear strain, spline collocation Introduction. Shallow shells are widely used in many fields of modern engineering, production, architecture, building, etc. nisotropic shells with various geometries, dimensions, and material properties are used as structural elements of various machines and mechanisms. The fundamentals of the theory of shallow shells are outlined in the monographs [1, 2, 5, 6]. Improved technologies and enhanced and sophisticated engineering designs require higher strength and reliability of critical components of many structures such as shallow shells. This is why it is important to develop efficient numerical and experimental methods for the analysis of the load-bearing capacity of shells, including the determination of resonant frequencies.In this connection, it is of interest to study the resonant modes and frequencies of shells of variable thickness with various shapes, material characteristics, and boundary conditions. The Timoshenko-Mindlin theory is not so popular as the classical Kirchhoff-Love theory in studying the free vibrations of anisotropic shallow shells. The fundamentals of the refined theory of shells are outlined in the monographs [1, 4, 6]. The stress-strain state of shallow shells was analyzed in [6, 12, 14, 19, etc.] based on a nonclassical approach. The free vibrations of plates were studied in [9, 10, 15, 16-18, etc.] using a refined approach.If shells are isotropic or orthotropic, have constant thickness, and are hinged, we can use the method of double trigonometric series. If, however, the shell is clamped, this method fails and a numerical method should be used. This class of problems is inadequately covered in the literature because of computational difficulties encountered in solving them.Recently, spline-functions have been widely used to solve problems of mechanics, including the analysis of the mechanical behavior of plates and shells. This is due to the following advantages of spline-approximations: (i) stability of splines against local perturbations (the local behavior of a spline at a point does not affect its overall behavior, unlike, for example, polynomial approximation); (ii) good convergence of spline-interpolation (unlike polynomial interpolation); and (iii) simplicity and convenience of numerical implementation of spline algorithms. When used in various variational, projective, and other discrete-continuous methods, spline-functions yield much better results than classical polynomials do, simplify the numerical implementation of these methods...