Abstract




 
   

IJE TRANSACTIONS C: Aspects Vol. 27, No. 3 (March 2014) 367-374   

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  CRACK DETECTION IN FUNCTIONALLY GRADED BEAMS USING CONJUGATE GRADIENT METHOD
 
Mo. Eftekhari, Ma. Eftekhari and M. Hosseini
 
( Received: December 24, 2012 – Accepted: August 22, 2013 )
 
 

Abstract    In this paper the conjugate gradient (CG) method is employed for identifying the parameters of crack in a functionally graded beam from natural frequency measurement. The crack is modeled as a massless rotational spring with sectional flexibility. By using the Euler-Bernoulli beam theory on two separate beams respectively and applying the compatibility requirements of the crack, the characteristic equation can be obtained as a function of natural frequency and location and depth of crack. In direct problem, the natural frequency is computed using analytical analysis. Moreover, the location and depth of crack are determined by measuring the three natural frequencies of beam in inverse problem. In this study, the CG method is utilized in inverse problem to determine the location and depth of crack. The obtained results show the efficiency of CG algorithm in terms of accuracy and the convergence speed.

 

Keywords    functionally graded beam, crack detection, conjugate gradient method

 

چکیده    چكيده در اين مقاله روش گراديان مزدوج برای مشخص کردن پارامترهای ترک در تير با خواص تابعی به کار رفته است که فرکانس‌های طبيعی از روش مستقيم بدست آمده‌اند. ترک به صورت فنر پيچشی بدون جرم با انعطاف‌پذيری مقطع مدل شده است. با استفاده از تئوری اويلر- برنولی روی دو تير جداگانه و اعمال شرايط سازگاری ترک، معادله مشخصه به صورت تابعی از فرکانس طبيعی تير و موقعيت و عمق ترک بدست می‌آيد. در حل مستقيم مساله ترک، فرکانس طبيعی از حل تحليلی معادله مشخصه بدست می‌آيد و موقعيت و عمق ترک با جايگذاری سه فرکانس‌های طبيعی در مساله معکوس مشخص می‌شود.در اين مطالعه از روش گراديان مزدوج برای محاسبه موقعيت و عمق ترک استفاده شده است. نتايج نشان دهنده کارايی روش گراديان مزدوج در تعيين دقت و سرعت همگرايی محاسبه موقعيت و عمق ترک است

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