Using the Fourier series for analysis of free vibrations of a cylindrical shell rotating on rollers

Abstract

Small free vibrations of an infinite cylindrical shell rotating on n rollers are considered. The shell is the model of a centrifugal concentrator intended for enrichment of ores. Displacements of the shell are searched as N-terms approximation of Fourier series on the circumference coordinate. The system of 2N linear differential equations for the factors of Fourier series is deduced. The solutions of these equations must satisfy the n equations of constraints. The system of 2N − n linear algebraic equations for calculating vibration mode shapes is obtained. Vibration frequencies !k, k = 1, 2, . . . , 2N − n are the roots of equation D(!2) = 0, where D is determinant of this system. The system of the 2N − n equations is equivalent to several independent systems with smaller that 2N −n number of unknowns. Hence the determinant D of the order 2N −n can be represented in the form of a product of determinants of the smaller order. In some cases, in particular in case N = n, frequencies are roots of the algebraic equations not above the second degree and can be found in explicit form.