Parametric resonance of shaft with variable slenderness ratio

Abstract

Parametric instability of a shaft occurs within a certain range of speed when the shaft parameters such as stiffness and mass are disturbed periodically. At high speed a shaft may carry high torque and as such the effect of torsion on the parametric instability need to be considered. The use of the Timoshenko’s beam theory in the parametric instability formulation of shaft in the past has seen this torsional effect being neglected. In this paper the torsional degree of freedom is added to the formulation for the parametric instability of shaft that is based on the mentioned Timoshenko’s theory. The focus is on the effect of slenderness ratio of shaft on the occurrence of parametric instability on the shaft. The finite element method is used to produce the Mathieu-Hill equation and upon applying the Bolotin’s method the parametric insta-bility equations of shaft can be determined. The formulation and its corresponding MATLAB source codes were successfully validated based on past results. It was found that the increase in the slenderness ratio has shifted the instability chart to the right by 20 %. Furthermore the additional torsional degree of freedom has significantly changed the instability chart of the shaft at high speed.