Parametric Instability of Shafts Based on the Nelson’s Beam Model Using Finite Element Method

Abstract

Dynamic instability condition may cause failure in machines and as such it is an important consideration in design process. In this work, parametric instability behavior was studied for a simple shaft system that was subjected to periodic axial loading. The beam is modelled based on the Nelson’s beam model that considered translational and rotary inertia, transverse shear deformation and the torsional effect. Finite element method has been used to develop the Mathieu-Hill equation of the dynamic instability of the shaft where with Nelson’s beam formulation, 10 dof per element was required. The Bolotin’s method was used to solve the Mathieu-Hill equation where then FEM codes were developed using MATLAB to establish the instability chart. The effect of the torsional motion on the stability chart was investigated for different conditions where numerical results and illustrative example are given. It is found that torsional motion has significant effect on instability region for the shaft system.