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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Studies on the static and dynamic deformation of isotropic and anisotropic elastic shell-like bodies of complex shape performed using classical and refined problem statements are reviewed. To solve two-dimensional boundary-value problems and eigenvalue problems, use is made of a nontraditional discrete-continuum approach based on the spline-approximation of the unknown functions of partial differential equations with variable coefficients. This enables reducing the original problem to a system of one-dimensional problems solved with the discrete-orthogonalization method. An analysis is made of numerical results on the distribution of stress and displacement fields and dynamic characteristics depending on the loading and boundary conditions, geometrical and mechanical parameters of elastic bodies. Emphasis is placed on the accuracy of the results Keywords: shell structures, static and dynamic problems, variable parameters, models, discrete-continuum methods Introduction.Many members of modern engineering structures have the form of intricately shaped shells fixed in various ways and subjected to various distributed and local loads. Shell elements are widely used to meet the requirements imposed by the severe operating conditions of machines, aircraft, transportation vehicles, industrial and civil facilities. The complication of the configuration of shell elements necessitates developing a theory and methods for solving static and dynamic problems for shells made of anisotropic inhomogeneous materials.A typical feature in the development of the theory of plates and shells is the relationship between the setting up of a mathematical model describing a given class of problems and the development of a method for solving them. For example, Kirchhoff's theory of thin plates [101] and the Kirchhoff-Love theory of thin shells [97] tend not only to provide a realistic description of the deformation of plates and shells, but also to make such models so simple as to solve a number of problems with available computational capability. An example of such relationship is the Donnell-Mushtari-Vlasov theory of shells [5,28,36]. Its basic equations were simplified so as to find solutions to some classes of problems over wide ranges of their characteristics, which is demonstrated by the names given to this theory: technical theory of shells, theory of shallow shells, theory of shells with highly variable stress state [2,8,9,16,20,23,24,29,37]. This relationship is even stronger nowadays when computers are widely used to solve problems in the theory of shells and mathematical models for certain classes of shells are set up so as to provide for all aspects and possibilities associated with the solution of problems [63].The solution of two-dimensional boundary-value problems of the statics and dynamics of plates, shells, and solids described by partial differential equations with variable coefficients involves severe computational difficulties. To solve them, use is sometimes made of approaches based on separation of variables and redu...

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Abstract. The present report proposes an efficient approach to solving within the framework of the classic and refined models the stress-strain problems of shallow shells as well as the problems on free vibrations. In accordance with the approach the initial system of partial differential equations is reduced to one-dimensional problems by using approximation of the solution in terms of basic splines in one coordinate. The boundary-value problems obtained and eigenvalue boundary-value problems for systems of ordinary differential high-order equations are solved by the stable numerical method of discrete ortogonalization.

Treatment of Class III is a current problem in orthodontics that requires constant improvement of its methods, development of new or modifications of known methods and techniques. We have developed and studied the modification of removable functionally-directing orthodontic appliances for treatment of Class III, which consists of a plastic base, vestibular arc, retaining clasps, ramp, which is connected with the base by means of two torsion springs. Its usage ensures a prolonged contact of ramp with the teeth. We studied two types of club-shaped springs (torsion springs): one spring, which create an amortization effect during the action of the ramp, but do not change its inclination angle and second one-spring that seek to increase the angle of the ramp inclination due to the disclosure of its curl.