Exact Solutions for Free Vibrations and Buckling of Double Tapered Columns With Elastic Foundation and Tip Mass

Abstract

The present study aims at the free vibration analysis of double tapered columns. Foundation is assumed to be elastic and the effects of self-weight and tip mass with significant moment of inertia are considered. The governing equation of motion is obtained using the Hamilton principle, based on both the Euler–Bernoulli and Timoshenko beam models. Applying the power series method of Frobenius, the base solutions of the governing equations are obtained in the form of a power series via general recursive relations. Applying the boundary conditions, the natural frequencies of the beam/column are obtained using both models. The obtained results are compared with literature and a very good agreement is achieved. Subsequently, comprehensive studies are performed to provide an insight into the variation of the natural frequencies and instability conditions of the beam with respect to the tip mass, self-weight, taper ratio, slenderness, and foundation stiffness and eventually some general conclusions are drawn.