Construction of an algorithm to analytically solve a problem on the free vibrations of a composite plate of variable thickness

Authors

  • Kirill Trapezon National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" Peremohy ave., 37, Kyiv, Ukraine, 03056, Ukraine https://orcid.org/0000-0001-5873-9519
  • Alexandr Trapezon G. S. Pisarenko Institute for Problems of Strength of the National Academy of Sciences of Ukraine Timiryazevs’ka str., 2, Kyiv, Ukraine, 01014, Ukraine https://orcid.org/0000-0002-8567-9854

DOI:

https://doi.org/10.15587/1729-4061.2020.191123

Keywords:

natural frequencies, vibration modes, analytical solution, annular plate, free vibrations, symmetry method

Abstract

The paper reports an algorithm to analytically solve one of the problems in the mechanics of elastic bodies, which is associated with studying the natural vibrations of a composite two-stage plate whose concave part is smoothly aligned with the part of a constant thickness. We have defined patterns for stating the boundary and transitional conditions, which should be taken into account when considering the natural vibrations of a two-stage plate.

The ratios have been obtained, which make it possible to study the distribution of deflections and determine the values of amplitudes of the curved vibrations of the plate. It was noted that the modes of vibrations are based on the symmetry and factorization methods that we had developed and refined earlier. Specifically, it has been found that the deflections can be explored through expressions that are derived through the sum of relevant solutions to two linear second-order differential equations with variable coefficients.

Based on the proposed approach, a system consisting of eight homogeneous algebraic equations has been defined, which allowed us to build a frequency equation for the plate rigidly fixed along the inner contour and free along the outer contour. We have determined the values for the plate’s natural frequencies for the first three modes of natural vibrations. Moreover, in order to verify and expand a set of plates of different configurations, the plates with two types of concave in their variable part have been considered. The new approaches and the ratios based on them could be useful for the further advancement of methods for solving similar problems in mathematical physics on natural values. A practical implementation is the problems about the vibrations of plates with variable thickness and of different modes

Author Biographies

Kirill Trapezon, National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" Peremohy ave., 37, Kyiv, Ukraine, 03056

PhD, Associate Professor

Department of Sound Engineering and Recording Information

Alexandr Trapezon, G. S. Pisarenko Institute for Problems of Strength of the National Academy of Sciences of Ukraine Timiryazevs’ka str., 2, Kyiv, Ukraine, 01014

Doctor of Technical Sciences, Leading Researcher

Laboratory No. 7.1

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Published

2020-02-29

How to Cite

Trapezon, K., & Trapezon, A. (2020). Construction of an algorithm to analytically solve a problem on the free vibrations of a composite plate of variable thickness. Eastern-European Journal of Enterprise Technologies, 1(7 (103), 26–33. https://doi.org/10.15587/1729-4061.2020.191123

Issue

Vol. 1 No. 7 (103) (2020): Applied mechanics

Section

Applied mechanics

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Copyright (c) 2020 Кирилл Трапезон, Александр Трапезон

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