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Mathematical Problems in Engineering
Volume 2016, Article ID 7314280, 12 pages
http://dx.doi.org/10.1155/2016/7314280
Research Article

Flexural Free Vibrations of Multistep Nonuniform Beams

Department of Road and Bridge, College of Transportation, Jilin University, Changchun 130022, China

Received 28 June 2016; Revised 20 September 2016; Accepted 11 October 2016

Academic Editor: Salvatore Caddemi

Copyright © 2016 Guojin Tan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents an exact approach to investigate the flexural free vibrations of multistep nonuniform beams. Firstly, one-step beam with moment of inertia and mass per unit length varying as and was studied. By using appropriate transformations, the differential equation for flexural free vibration of one-step beam with variable cross section is reduced to a four-order differential equation with constant coefficients. According to different types of roots for the characteristic equation of four-order differential equation with constant coefficients, two kinds of modal shape functions are obtained, and the general solutions for flexural free vibration of one-step beam with variable cross section are presented. An exact approach to solve the natural frequencies and modal shapes of multistep beam with variable cross section is presented by using transfer matrix method, the exact general solutions of one-step beam, and iterative method. Numerical examples reveal that the calculated frequencies and modal shapes are in good agreement with the finite element method (FEM), which demonstrates the solutions of present method are exact ones.