Eigenvalues in problem of free vibrations of rods with variable cross-section

The article considers free longitudinal vibrations of a steel rod of variable cross-section along the length. The left end of the rod is pinched, and the mass and the coil spring are concentrated at the right end. The basic equation of the mathematical model of vibrations is compiled using the d...

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Главные авторы:	Kulterbaev, K. P., Культербаев, Х. П.
Формат:	Статья
Язык:	English
Опубликовано:	American Institute of Physics 2024
Темы:	Variable cross-section Algebraic equations
Online-ссылка:	https://dspace.ncfu.ru/handle/123456789/29340
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id	ir-123456789-29340
record_format	dspace
spelling	ir-123456789-293402024-12-09T13:23:33Z Eigenvalues in problem of free vibrations of rods with variable cross-section Kulterbaev, K. P. Культербаев, Х. П. Variable cross-section Algebraic equations The article considers free longitudinal vibrations of a steel rod of variable cross-section along the length. The left end of the rod is pinched, and the mass and the coil spring are concentrated at the right end. The basic equation of the mathematical model of vibrations is compiled using the d'Alembert principle. The result is a model including a hyperbolic partial differential equation and boundary conditions corresponding to the support conditions for the left and right ends of the rod. Free vibrations are undamped and harmonic without initial conditions. The purpose of solving the problem is to determine the frequencies of free vibrations. The use of the finite difference method for solving the problem is substantiated. The domain of continuous variation of the displacement function argument along the rod axis is replaced by a uniform grid function. Therefore, the mathematical model turns into a system of algebraic equations. The desired vibration frequencies are determined as eigenvalues of a square matrix. At the final stage of determining the free vibrations frequencies, a numerical-graphical method implemented in the environment of the Matlab computer complex is used. A specific example is given. The analysis results in the conclusions important for practical applications. 2024-12-09T13:22:57Z 2024-12-09T13:22:57Z 2024 Статья Lafisheva, M.M., Kulterbaev, K.P., Baragunova, L.A., Shogenova, M.M., Tseeva, F.M. Eigenvalues in problem of free vibrations of rods with variable cross-section // AIP Conference Proceedings. - 2024. - 3183 (1). - статья № 050005. - DOI: 10.1063/5.0229554 https://dspace.ncfu.ru/handle/123456789/29340 en AIP Conference Proceedings application/pdf American Institute of Physics
institution	СКФУ
collection	Репозиторий
language	English
topic	Variable cross-section Algebraic equations
spellingShingle	Variable cross-section Algebraic equations Kulterbaev, K. P. Культербаев, Х. П. Eigenvalues in problem of free vibrations of rods with variable cross-section
description	The article considers free longitudinal vibrations of a steel rod of variable cross-section along the length. The left end of the rod is pinched, and the mass and the coil spring are concentrated at the right end. The basic equation of the mathematical model of vibrations is compiled using the d'Alembert principle. The result is a model including a hyperbolic partial differential equation and boundary conditions corresponding to the support conditions for the left and right ends of the rod. Free vibrations are undamped and harmonic without initial conditions. The purpose of solving the problem is to determine the frequencies of free vibrations. The use of the finite difference method for solving the problem is substantiated. The domain of continuous variation of the displacement function argument along the rod axis is replaced by a uniform grid function. Therefore, the mathematical model turns into a system of algebraic equations. The desired vibration frequencies are determined as eigenvalues of a square matrix. At the final stage of determining the free vibrations frequencies, a numerical-graphical method implemented in the environment of the Matlab computer complex is used. A specific example is given. The analysis results in the conclusions important for practical applications.
format	Статья
author	Kulterbaev, K. P. Культербаев, Х. П.
author_facet	Kulterbaev, K. P. Культербаев, Х. П.
author_sort	Kulterbaev, K. P.
title	Eigenvalues in problem of free vibrations of rods with variable cross-section
title_short	Eigenvalues in problem of free vibrations of rods with variable cross-section
title_full	Eigenvalues in problem of free vibrations of rods with variable cross-section
title_fullStr	Eigenvalues in problem of free vibrations of rods with variable cross-section
title_full_unstemmed	Eigenvalues in problem of free vibrations of rods with variable cross-section
title_sort	eigenvalues in problem of free vibrations of rods with variable cross-section
publisher	American Institute of Physics
publishDate	2024
url	https://dspace.ncfu.ru/handle/123456789/29340
work_keys_str_mv	AT kulterbaevkp eigenvaluesinproblemoffreevibrationsofrodswithvariablecrosssection AT kulʹterbaevhp eigenvaluesinproblemoffreevibrationsofrodswithvariablecrosssection
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Eigenvalues in problem of free vibrations of rods with variable cross-section

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