Solving Equations of Free Vibration for a Cylindrical Shell Rotating on Rollers by the Fourier Method

Research output

Abstract

The small free vibrations of an infinite circular cylindrical shell rotating about its axis at a constant angular velocity are considered. The shell is supported on n absolutely rigid cylindrical rollers equispaced on its circle. The roller-supported shell is a model of an ore benefication centrifugal concentrator with a floating bed. The set of linear differential equations of vibrations is sought in the form of a truncated Fourier series containing N terms along the circumferential coordinate. A system of 2N–n linear homogeneous algebraic equations with 2N–n unknowns is derived for the approximate estimation of vibration frequencies and mode shapes. The frequencies ωk, k = 1, 2, …, 2N–n, are positive roots of the (2N–n)th-order algebraic equation D(ω2) = 0, where D is the determinant of this set. It is shown that the system of 2N–n equations is equivalent to several independent systems with a smaller number of unknowns. As a consequence, the (2N–n)th-order determinant D can be written as a product of lower-order determinants. In particular, the frequencies at N = n are the roots of algebraic equations of an order is lower than 2 and can be found in an explicit form. Some frequency estimation algorithms have been developed for the case of N > n. When N increases, the number of found frequencies also grows, and the frequencies determined at N = n are refined. However, in most cases, the vibration frequencies can not be found for N > n in an explicit form.

Original languageEnglish
Pages (from-to)182-191
Number of pages10
JournalVestnik St. Petersburg University: Mathematics
Volume51
Issue number2
DOIs
Publication statusPublished - 1 Apr 2018

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Fourier Method
Cylindrical Shell
Free Vibration
Rotating
Algebraic Equation
Determinant
Vibration
Shell
Roots
Frequency Estimation
Unknown
Concentrator
Mode Shape
Angular velocity
Estimation Algorithms
Linear differential equation
Fourier series
System of equations
Circle
Term

Scopus subject areas

  • Mathematics(all)

Cite this

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title = "Solving Equations of Free Vibration for a Cylindrical Shell Rotating on Rollers by the Fourier Method",
abstract = "The small free vibrations of an infinite circular cylindrical shell rotating about its axis at a constant angular velocity are considered. The shell is supported on n absolutely rigid cylindrical rollers equispaced on its circle. The roller-supported shell is a model of an ore benefication centrifugal concentrator with a floating bed. The set of linear differential equations of vibrations is sought in the form of a truncated Fourier series containing N terms along the circumferential coordinate. A system of 2N–n linear homogeneous algebraic equations with 2N–n unknowns is derived for the approximate estimation of vibration frequencies and mode shapes. The frequencies ωk, k = 1, 2, …, 2N–n, are positive roots of the (2N–n)th-order algebraic equation D(ω2) = 0, where D is the determinant of this set. It is shown that the system of 2N–n equations is equivalent to several independent systems with a smaller number of unknowns. As a consequence, the (2N–n)th-order determinant D can be written as a product of lower-order determinants. In particular, the frequencies at N = n are the roots of algebraic equations of an order is lower than 2 and can be found in an explicit form. Some frequency estimation algorithms have been developed for the case of N > n. When N increases, the number of found frequencies also grows, and the frequencies determined at N = n are refined. However, in most cases, the vibration frequencies can not be found for N > n in an explicit form.",
keywords = "Fourier series, free vibrations, rotating cylindrical shell, system of linear algebraic equations",
author = "Filippov, {S. B.}",
year = "2018",
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AB - The small free vibrations of an infinite circular cylindrical shell rotating about its axis at a constant angular velocity are considered. The shell is supported on n absolutely rigid cylindrical rollers equispaced on its circle. The roller-supported shell is a model of an ore benefication centrifugal concentrator with a floating bed. The set of linear differential equations of vibrations is sought in the form of a truncated Fourier series containing N terms along the circumferential coordinate. A system of 2N–n linear homogeneous algebraic equations with 2N–n unknowns is derived for the approximate estimation of vibration frequencies and mode shapes. The frequencies ωk, k = 1, 2, …, 2N–n, are positive roots of the (2N–n)th-order algebraic equation D(ω2) = 0, where D is the determinant of this set. It is shown that the system of 2N–n equations is equivalent to several independent systems with a smaller number of unknowns. As a consequence, the (2N–n)th-order determinant D can be written as a product of lower-order determinants. In particular, the frequencies at N = n are the roots of algebraic equations of an order is lower than 2 and can be found in an explicit form. Some frequency estimation algorithms have been developed for the case of N > n. When N increases, the number of found frequencies also grows, and the frequencies determined at N = n are refined. However, in most cases, the vibration frequencies can not be found for N > n in an explicit form.

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