Free Vibration Analysis of Non-uniform Rayleigh Beams on Variable Winkler Elastic Foundation using Differential Transform Method

This study examines the effect of variable Winkler foundation on the natural frequencies of a prestressed nonuniform Rayleigh beam. In this work, the elastic coefficients of the foundations are assumed to vary along the length direction of the beam. A semi-analytical approach known as Differential Transform Method (DTM) is applied to the non-dimensional form of the governing equations of motion of the prestressed non-uniform Rayleigh beam and a set of recursive algebraic equations are obtained. Evaluating these derived equations and using some computer codes written and implemented in MAPLE 18, the non-dimensional frequencies and the associated mode shapes of the beam are obtained. The effects of variable Winkler foundation variations and axial force for various values of the slenderness ratio on the non-dimensional frequencies are investigated. The clamped-clamped and simply supported boundary conditions are considered to illustrate the accuracy and efficiency of this method. Finally, the results obtained are validated and are found to compare favorably well with those in the open literature.