Determination of Eigenforms and Frequencies of Transverse Vibrations of a Rod of Variable Cross Section in the Field of Centrifugal Forces

Abstract

The authors of the article have developed a unified method for determining the forms and frequencies of free transverse vibrations of a direct rod of variable cross section, taking into account tensile forces caused by the rotation of the rod. The technique is based on finite element approximations where the rod is represented as a set of four-degree bendable elements. The Kirchhoff–Love hypothesis is used while calculating. To obtain the equations of motion of finite elements, the general dynamic equation is applied. The mass matrix, the physical stiffness matrix, and the geometrical stiffness matrix of the final element are obtained taking into account the linear law of variation of the linear mass, flexural rigidity, and tensile centrifugal force along the length of the element. To get the equations of free vibrations of a finite element rod model, the authors have used the general dynamic equation. They have carried out the approbation of the developed technique with the determination of several low natural forms and frequencies of transverse vibrations of a rod of variable thickness rotating around a fixed axis. To determine these forms and frequencies, the iteration method in the subspace is used. This method allows calculating the lower forms and frequencies of natural vibrations of nodes, units, and structures operating in the field of centrifugal forces. The described algorithm is implemented as a program in the MATLAB package.