TY - JOUR

T1 - Solving Equations of Free Vibration for a Cylindrical Shell Rotating on Rollers by the Fourier Method

AU - Filippov, S. B.

PY - 2018/4/1

Y1 - 2018/4/1

N2 - 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.

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.

KW - Fourier series

KW - free vibrations

KW - rotating cylindrical shell

KW - system of linear algebraic equations

UR - http://www.scopus.com/inward/record.url?scp=85048700144&partnerID=8YFLogxK

U2 - 10.3103/S1063454118020036

DO - 10.3103/S1063454118020036

M3 - Article

AN - SCOPUS:85048700144

VL - 51

SP - 182

EP - 191

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 2

ER -