A technique for applying the symmetry method to solve a problem of torsional vibrations of disks of variable thickness
DOI:
https://doi.org/10.15587/1729-4061.2025.323561Keywords:
differential equation, disk of variable thickness, torsional vibrations, natural frequencies, method of symmetriesAbstract
The object of this study is a disk of variable thickness. A solution to the problem of natural torsional vibrations of disks of variable thickness was sought. An algorithm to solve the problem for an arbitrary number of different disk profiles has been constructed. The law of change in disk thickness H(ρ), which contains three arbitrary constants α, С, С1, has been considered. The choice of the disk profile configuration is controlled by changing the values of these three constants.
Exact solutions to the problem in elementary functions are known only in two cases, when H(ρ)=1/ρ3 or H(ρ)=ρ-3eαρ, where ρ is the relative radial coordinate and α is an arbitrary constant. These cases are not sufficient for generalizing conclusions about the behavior of disks during their oscillations.
For the case of a disk that is rigidly fixed along its inner diameter and with a free outer edge, the corresponding relations were derived. They made it possible to calculate natural numbers and study the distribution of angular displacements of the disk. These numerical parameters, along with the frequency indices, are a convenient technique for evaluating the resonant properties of the disk for practice.
A comparison of torsional and radial vibrations of the disk with the chosen law of thickness change was performed. To study torsional vibrations, approximation approaches of thickness change functions were used. It was found that the relative discrepancy between the values of these functions at a certain interval did not exceed 2.2 %. It was found that the differences in the eigenfrequencies of torsional vibrations for the disk of the chosen configuration were significantly smaller than in the case of radial vibrations.
A practical algorithm for applying the method used is presented, which could prove useful for further research based on similar analytical approaches.
The method makes it possible to choose the desired disk configuration for various practical purposes. Owing to this feature, it is possible to provide the required distribution of cyclic stresses, resonant frequencies, and amplitudes for the disk
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