Abstract




 
   

IJE TRANSACTIONS A: Basics Vol. 30, No. 10 (October 2017) 1565-1572    Article in Press

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  VARIATIONAL ITERATION METHOD FOR FREE VIBRATION ANALYSIS OF A TIMOSHENKO BEAM UNDER VARIOUS BOUNDARY CONDITIONS
 
K. Torabi, M. Ghassabi, M. Heidari-Rarani, D and Sharifi
 
( Received: April 28, 2017 – Accepted in Revised Form: July 07, 2017 )
 
 

Abstract    In this paper, a relatively new method, namely variational iteration method (VIM), is developed for free vibration analysis of a Timoshenko beam with different boundary conditions. In the VIM, an appropriate Lagrange multiplier is first chosen according to order of the governing differential equation of the boundary value problem, and then an iteration process is used till the desired accuracy is achieved. Solution of VIM for natural frequencies and mode shapes of a Timoshenko beam is compared to the available exact closed-form solution and numerical results of differential quadrature method (DQM). The accuracy of VIM is approximately the same as exact solution and much better than the DQM for solving the free vibration of a Timoshenko beam. Also, convergence speed and simplicity of this method is more than the other two methods because it works with polynomial at the first iteration. Thus, VIM can be used for solving the complicate engineering problems which do not have analytical solution.

 

Keywords    Variational iteration method (VIM), boundary value problem, free vibration, Timoshenko beam.

 

چکیده    در این مقاله از یک روش نسبتاً جدید به نام روش تکرار تغییرات (VIM) برای تحلیل ارتعاشات آزاد تیر تیموشنکو با شرایط مرزی مختلف استفاده شده است. در VIM، ابتدا بر اساس مرتبه معادله دیفرانسیل مسئله شرط مرزی داده شده، ضریب لاگرانژ مناسب انتخاب می شود و سپس فرآیند تکرار تا رسیدن به دقت مطلوب ادامه دارد. پاسخ­های VIM برای فرکانس های طبیعی و شکل مودهای یک تیر تیموشنکو با پاسخ حل دقیق و نتایج عددی روش DQM مقایسه شده است. دقت VIM تقریباً همان دقت حل دقیق بوده و بسیار از حل DQM برای حل ارتعاشات آزاد تیر تیموشنکو دقیق­تر است. از طرفی، سرعت همگرایی و سادگی این روش از دو روش دیگر بسیار بهتر است چرا که از یک چند جمله­ایی در تکرار اول استفاده می شود. بنابراین VIM می­تواند برای مسائل پیچیده مهندسی که نمی توان از روش تحلیلی بهره برد، استفاده شود.

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