Mathematical model of the problem of vibration of thin-walled structural elements has been constructed based on Kirchhoff-Love theory. The problem is reduced, using the Bubnov-Galerkin method, to the solution of a set of nonlinear integro-differential Volterra type equations with weakly-singular kernels of relaxation. A numerical method based on the use of quadrature formulae being used for their solution. The influence of rheological parameters of the material on the values of critical velocity and amplitude-frequency characteristics of viscoelastic thin-walled structural elements is analyzed. It is shown that tacking account viscoelastic properties of the material of thin-walled structures lead to a decrease in the critical rate of gas flow.

The paper is devoted to the study of nonlinear problems of the dynamics of thin-walled structures considering hereditary and inhomogeneous properties of the material. To describe the processes of strain in viscoelastic materials, the Boltzmann-Volterra integral model with weakly singular hereditary kernels was used. Using the Bubnov-Galerkin method, the problem under consideration was reduced to solving the systems of nonlinear integrodifferential equations considering hereditary and inhomogeneous properties of the material and the radius of curvature of the structure. A computational algorithm was developed based on the elimination of the features of integrodifferential equations with weakly singular kernels, followed by the use of quadrature formulas. The results of calculating a thin-walled structure with hereditary and inhomogeneous properties of the material streamlined by a gas flow are presented. Solutions are obtained in the form of graphs.

The paper presents the results of an oscillation process study of thin-walled structures viscoelastic elements, taking into account the static pressure drop. When studying the oscillations of thin-walled structure elements in a gas flow, a model in the form of a cylindrical panel was used. To describe the viscoelastic properties, the hereditary Boltzmann-Volterra theory of viscoelasticity was applied. When realizing the physicomechanical properties of the object material, the systems of integro-differential equations (IDE) in partial derivatives with corresponding initial and boundary conditions are the mathematical model of the problems under consideration. The obtained nonlinear partial differential equations using the Bubnov-Galerkin method were reduced to the solution of nonlinear ordinary differential equations with constant or variable coefficients with respect to the time function. The integration of the equations obtained using the polynomial approximation of deflections was carried out numerically. Based on this method, an algorithm for the numerical solution of the problem was developed fit for all viscoelastic elements of thin-walled structures of panel type.

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