We propose an algorithm for the evaluation of frequencies and forms of natural vibrations of a thin-walled circular cylindrical shell reinforced by a transverse annular elastic rib of small width with regard for the presence of discontinuities of the first kind in the force factors acting on the line of contact of the rib with the shell. To construct approximate solutions of the analyzed spectral problem, we use the variational method in combination with the partition of the domain of definition of the required functions into regular subdomains in each of which the displacements, forces, and moments have the properties of continuity and differentiability. The Ritz method proposed for the solution of the analyzed problem guarantees convergence in the uniform metric for displacements and the force factors of elastic shell.

We consider a system of differential equations that describes free oscillations of a thin-walled conic shell of revolution with vertex. On the basis of the analytic theory of systems of differential equations with small parameter at the highest derivative and equations with regular singular point, we establish the formal structure of regular integrals of the original equations.

We develop a variational method for the solution of the spectral problem of free axially symmetric vibration of a thin-walled conic shell with vertex. The procedure of construction of systems of basis functions for the approximation of displacements of the shell is based on the account of the formal structure of the fundamental system of integrals of the original equations established as a result of the asymptotic integration of a singularly perturbed system of equations with regular singular point. We present examples of numerical analyses of specific conic shells illustrating the efficiency of the proposed algorithm for the solution of the problem.

We propose a variational method for the construction of approximate solutions of the spectral problem of vibrations of two conjugate shells of revolution that are not axially symmetric. The solution of the problem is based on the decomposition of the domain of integration of equations of the theory of shells performed by using the variational method. We construct a generalized functional of displacements of the shell for which the conditions of conjugation of solutions on the common boundary of the introduced subdomains are natural boundary conditions. For a shell formed by a truncated cone and a cylinder, the efficiency of the proposed approach is analyzed and the results of numerical calculations are compared with the available data obtained by the other authors.

УДК 539.3:534.13 Розглядається система диференцiальних рiвнянь, яка описує вiльнi коливання тонкостiнної конiчної оболонки обертання з вершиною. Виходячи з аналiтичної теорiї систем диференцiальних рiвнянь з малим параметром при старшiй похiднiй i рiвнянь з регулярною особливою точкою встановлена формальна структура регулярних iнтегралiв початкових рiвнянь

We propose an algorithm for the numerical analyses of vibrations of elastic shells of revolution partially filled with ideal incompressible liquids. In the solution of this problem, the wave motions of liquid on its free surface are taken into account. The solution of the problem of hydroelasticity is based on the application of the method of decomposition of the domain of integration of equations of the theory of shells with the use of the variational statement of the problem and on the approximate construction of the operator inverse to the operator of the hydrodynamic part of the problem. We construct a generalized functional of displacements of the shell for which the role of conditions of matching for the solutions in different subdomains is played by the natural boundary conditions. The obtained numerical results are compared with the existing exact solutions of the analyzed problem for a shell in the form of a straight circular cylinder.

We propose systems of coordinate functions that can be used in the Ritz method aimed at finding the natural modes and eigenfrequencies of nonaxially symmetric oscillations of thin-walled dome-shaped shells of revolution. The basis functions are constructed with regard for the specific features of the spectral problem, which guarantees the uniform convergence of the process of calculations. As an example, we find the dynamic characteristics for a shell in the form of a spherical dome.

We propose an algorithm for finding the frequencies and modes of natural vibrations of the shells of revolution partially filled with liquid. The problem of perturbed motion of a liquid is solved under the assumption that its free surface remains flat and perpendicular to the axis of the shell. The solution is based on the use of the method of decomposition of the domain of integration of the equations of the theory of shells in combination with the variational method and the approximate construction of the inverse operator for the hydrodynamic part of the problem. We construct a generalized functional with respect to displacements of the shell for which the conditions of conjugation of the solutions in subdomains are included in the natural boundary conditions. The obtained numerical results are compared with the available exact solutions of the problem under consideration with regard for the wave motions of liquid in a shell in the form of circular cylinder.

We develop a variation method for the solution of the spectral problems for free oscillations of liquids in axially symmetric vessels of complex geometry. The problem is formulated in terms of the conjugation method. We obtain a generalized functional for which the conjugation conditions on the adjacent parts of the introduced subdomains are natural boundary-value conditions. We use the Trefts method for the reduction of the original problem to a problem of solution of an algebraic problem of low dimension. The results of calculations confirm the efficiency of the proposed method.

This paper generalizes earlier authors’ results on the analytical approximation of the singularly perturbed boundary problem describing the eigenoscillations of a thin-walled axisymmetric shell. The asymptotic behavior of the eigenmodes at the clamped ends is studied, and a set of trial functions capturing this behavior is constructed to be used in the Ritz method. Illustrative numerical examples demonstrate a fast convergence so that the eigenmodes are accurately approximated in a uniform metric together with their second-, third-, and fourth-order derivatives. The numerical results are validated by comparing them with an asymptotic eigensolution and computations done by the ANSYS codes based on the finite-element method.

Employing the virtual work variational principle and the linear multimodal method for the liquid sloshing in an axisymmetric tank, we study coupled eigenoscillations of a tower and an elevated tank partially filled by a liquid. An emphasis is placed on the case of an upright circular cylindrical tank. Theoretical results are compared with known experimental data.

We consider the problem of free oscillations of an ideal fluid in a container that has the form of a right circular cylinder with arbitrary axisymmetric bottom in the case where the unperturbed free surface of the fluid is covered by an elastic membrane or plate. Using the expansion in eigenfunctions of an auxiliary spectral problem with a parameter in boundary conditions and the method of decomposition of the domain of meridional cross-section of a container, we obtain an analytic solution of the problem. Individual examples of mechanical systems are considered, for which we construct solutions by using the proposed algorithm, analyze these solutions, and compute the frequencies and forms of oscillations.

Eigenfrequencies and eigenmodes of composite mechanical systems consisting of a thin-walled cylindrical shell and elastic beams [beam–shell–beam (BSB), beam–beam–beam (BBB), etc. systems] are described by using semi-analytical methods. The methods are less universal comparing with the Finite Element Method, but they are very accurate and CPU-efficient, and they could have advantages in studying multicomponent structures. A comparative analysis of eigenfrequencies and eigenmodes of the considered composite systems versus characteristic geometric dimensions is presented.

Based on the linear modal sloshing theory and a variational statement, a nonclassical “hybrid” boundary problem is derived to describe coupled dynamics of a tower with an elevated tank on the tower top. Mathematically, the problem couples the generalized Euler-Bernoulli beam equation and an infinite-dimensional system of linear ordinary differential equations. The coupled eigenoscillations of the whole composite structure are analyzed. c © Gavrilyuk, Hermann, V.Trotsenko, Yu.Trotsenko, Timokha

We propose approximate solutions of two-dimensional hydroelastic problems that describe free oscillations of an ideal fluid in a horizontal long cylindrical container with arbitrary symmetric cross section. The free surface of the fluid is covered by a plane membrane or an elastic plate. Using specific examples, we analyze the obtained solutions and the results of computation of frequencies and forms of oscillations of the mechanical system under consideration.

In this paper a numerical method is proposed to compute the eigenoscillations of a thin-walled non-closed shell of revolution. The method is based on the well-known Ritz method. The use of special coordinate functions which are adapted to the boundary layer behaviour at the clamped ends guarantees a uniform convergence to the natural modes and their (up to fourth order) derivatives. It is shown that the convergence of the new method does not significantly depend on the thickness of the shell. c © Gavrilyuk, Hermann, V.Trotsenko, Yu.Trotsenko, Timokha

We consider a mechanical system consisting of a circular cylindrical shell and perfectly rigid body attached to one of the shell ends. Starting from the principle of virtual works, we construct a mathematical model of the equilibrium state of our system subjected to stresses of general form. A boundary eigenvalue problem describes free vibrations of the “body – shell” system, and its approximate solution is determined. We construct the exact solution of the above problem by replacing the shell with an equivalent Timoshenko beam. The effect of the rigid body on the system vibrations is estimated, and the accuracy of the beam approximation to shell bending vibrations is studied.