DYNAMIC STABILITY AND FREE VIBRATIONS OF A SHAFT MOUNTED ON VISCOELASTIC SUPPORTS: PROBLEM FORMULATION AND SOLUTION METHODOLOGY

Abstract

This paper addresses the dynamic stability and free vibrations of a mechanical system with a finite number of degrees of freedom, consisting of a deformable shaft mounted on viscoelastic supports. The mathematical formulation of the problem is based on the principle of possible displacements, from which a general variational equation is derived. The rheological properties of the deformable supports and the shaft are described using the linear hereditary Boltzmann-Volterra relation together with the weakly singular Koltunov-Rzhanitsyn kernel. Owing to the smallness of the integral terms in the constitutive relation, the "freezing" method is applied, and the transcendental equation determining the free-vibration frequencies of the system is solved numerically by the Müller method. The dynamic stability of the rotating shaft-support system is assessed using the D-partition method. The obtained results are compared with known solutions by A.S. Kelzon and V.I. Yakovlev, the discrepancy not exceeding 15%.