Presenting an analytical relation to determine frequency and mode shape of stepped hinged beam resting on elastic foundation

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Head of Department-Islamic Azad University- Arak Branch

Abstract

In stepped beams for finding natural frequency, the beam is divided into two segments in step location and using boundary conditions and continuity conditions between two segments eigenvalue problem has been computed. This procedure needs the numerical solution to determine frequency and cannot show the affects of various parameters on the frequency explicitly. In this paper, a polynomial function added to fundamental mode shape of hinged-hinged beam and obtained function considered as fundamental mode shape of stepped beam. then, the unknown coefficients of polynomial are determined by using the boundary conditions and continuity conditions in step location, and an analytical expression has been derived for fundamental frequency by using Rayleigh method which shows effects of different parameters in explicit form. the accuracy of derived formulation is confirmed by comparing with exact analytical solution of stepped hinged beam on elastic foundation for different values of step location and ratio. The derived analytical relation in this paper is a simple and accurate relation for determining the fundamental frequency of stepped hinged beam on elastic foundation.

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Journal of Solid and Fluid Mechanics
Volume 4, Issue 3 - Serial Number 3
September and October 2014
Pages 57-64

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