CALCULATION OF FREE AXISYMMETRIC VIBRATIONS OF CIRCULAR PLATES SUPPORTED BY A POWER-VARIABLE ELASTIC WINKLER BASE
DOI:
https://doi.org/10.31650/2786-6696-2025-12-51-64Keywords:
circular plate, inhomogeneous base, Winkler's hypothesis, variable bedding coefficient, axisymmetric vibrations, analytical solution.Abstract
This publication is about further development of analytical methods for calculating free vibrations of circular plates resting on an inhomogeneous continuous elastic Winkler foundation. The inhomogeneity of the elastic foundation is described by a variable subgrade modulus. The case of subgrade modulus represented by a power function is considered. The fundamental functions of the corresponding vibration equation for circular solid plates are written down in an explicit closed form. These functions are dimensionless and are represented by absolutely and uniformly convergent double power series. In turn, these functions are used to express the formulas for the dynamic parameters of the plate state, such as deflection, angle of rotation, radial and circular bending moments, and shear force. An analytical representation for the frequency of free vibrations of the plate is obtained, which establishes its dependence on the dimensionless frequency and other mechanical parameters of the system. The dimensionless frequency, in turn, is determined from the frequency equations, which are obtained after the implementation of the specified boundary conditions.
The practical application of the obtained solutions is demonstrated by an example. A concrete slab with a rigidly fixed contour resting on a power-variable elastic foundation is examined. The first five frequencies of axisymmetric vibrations are calculated by the authors' method (AM). The corresponding first five modes are also presented in graphical format. The numerical values obtained by AM are treated as accurate, since the used calculation method is based on the exact solution of the corresponding differential equation. The availability of such solutions makes it possible to evaluate the accuracy of calculations obtained using various approximate methods by comparison. For the purpose of such a comparison, the paper presents the results of the calculation obtained by the finite element method (FEM). The relative error of the FEM in the calculation of this structure is determined.
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