Table of Contents Author Guidelines Submit a Manuscript
Shock and Vibration
Volume 18, Issue 4, Pages 555-562

Exact Longitudinal Vibration Characteristics of Rods with Variable Cross-Sections

Bulent Yardimoglu and Levent Aydin

Department of Mechanical Engineering, Izmir Institute of Technology, Urla, 35430 Izmir, Turkey

Received 7 January 2010; Revised 27 February 2010

Copyright © 2011 Hindawi Publishing Corporation. 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.


Longitudinal natural vibration frequencies of rods (or bars) with variable cross-sections are obtained from the exact solutions of differential equation of motion based on transformation method. For the rods having cross-section variations as power of the sinusoidal functions of ax+b, the differential equation is reduced to associated Legendre equation by using the appropriate transformations. Frequency equations of rods with certain cross-section area variations are found from the general solution of this equation for different boundary conditions. The present solutions are benchmarked by the solutions available in the literature for the special case of present cross-sectional variations. Moreover, the effects of cross-sectional area variations of rods on natural characteristics are studied with numerical examples.