Stability investigations on greenhill shafts using variational iteration method

Abstract

This paper investigates the stability of Greenhill’s shafts subjected to non-conservative torques and fixed axial compressive load using. Variational Iteration Method (VIM). The effectiveness of VIM in solving a differential equation primarily depends on the Lagrangian multiplier used in the recurrence formula for iterations. In this paper, exact Lagrangian multipliers are proposed for the first time to solve the governing differential equations of Greenhill’s shafts using VIM. Using these exact multipliers, the divergence and flutter loads of Greenhill’s shafts are determined. The interaction between conservative fixed loads and the non-conservative axial moments are also studied. Comparisons are drawn between the convergence of the numerical results obtained using the proposed exact Lagrangian multipliers and the approximate Lagrangian multipliers. The paper also focusses on the Nicolai’s variant of a Greenhill’s cantilever shaft subjected to axial torque. Parametric studies are conducted using VIM to study the effect of damping and imperfections on the behavior of Nicolai’s shaft. The results obtained from VIM are used to create stability maps of shafts subjected to tangential torque for different levels of damping.