Transverse vibrations of a rod with variable cross sections and calculation of its eigenfrequencies by the method of splines

Abstract

Discusses own transverse vibrations of a straight rod with a cross section of rectangular shape having a constant height and variable width, changing for exponen social law. Analytical method the obtained values of natural frequencies for various functions of changes in the cross section of a core and various methods of its fastening. Found that some analytical results are markedly different from those previously obtained by other authors. Also to analyze the spectrum of frequencies of own fluctuations of a rod of variable cross section applied numerical method ? method of splines of degree 5 defect 1. A detailed study of the error of the method of splines in comparison with the analytical solution showed that all the considered tasks, this method is characterized by second order of convergence and ensures the accuracy of the calculation chaquencies of oscillations for the rst ten natural modes of at least six significant digits. This allows us to consider the proposed method of splines is very effective for the analysis of se processes transverse vibrations of rods with variable cross sections in cases where the exact analytical solution is not known.