Abstract
In this paper, the integro-differential equations of natural oscillations of a viscoelastic ribbed truncated conical shell are obtained based on the Lagrange variational equation. The general research methodology is based on the variational principles of mechanics and variational methods. Geometrically nonlinear mathematical models of the deformation of ribbed conical shells are obtained, considering such factors as the discrete introduction of edges. Based on the finite element method, a method for solving and an algorithm for the equations of natural oscillations of a viscoelastic ribbed truncated conical shell with articulated and freely supported edges is developed. The problem is reduced to solving homogeneous algebraic equations with complex coefficients of large order. For a solution to exist, the main determinant of a system of algebraic equations must be zero. From this condition, we obtain a frequency equation with complex output parameters. The study of natural vibrations of viscoelastic panels of truncated conical shells is carried out, and some characteristic features are revealed. The complex roots of the frequency equation are determined by the Muller method. At each iteration of the Muller method, the Gauss method is used with the main element selection. As the number of edges increases, the real and imaginary parts of the eigenfrequencies increase, respectively.
Highlights
Conical shell structures are widely used in rocketry, aircraft construction, shipbuilding, and construction
A considerable number of theoretical and experimental works have been devoted to the study of the natural oscillations of circular cones
There are works in which the dependences for determining the resonant frequencies [1] and the vibration forms of truncated conical panels [2,3] are obtained by theoretical and experimental methods. Another method is mainly used to study shells, which allows us to move from the stability equations of conical shells to the corresponding equations for cylindrical shells with a circular cross-section
Summary
Conical shell structures are widely used in rocketry, aircraft construction, shipbuilding, and construction. There are works in which the dependences for determining the resonant frequencies [1] and the vibration forms of truncated conical panels [2,3] are obtained by theoretical and experimental methods. The literature analysis shows that the existing optimal shell designs for the given geometric and rheological parameters cannot be implemented in practice; the level of research remains only theoretical In this regard, despite the long history of solving the problem, the determination of the resonant frequency of natural oscillations, taking into account the structural properties of ribbed shells, remains relevant. The aim of this work is to develop a method, algorithm, and program for finding resonant frequencies and vibration forms for circular ribbed viscoelastic conical shells under various boundary conditions
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