IJE TRANSACTIONS C: Aspects Vol. 27, No. 3 (March 2014) 385-394   

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A. Mirzabeigy
( Received: June 30, 2013 – Accepted: September 14, 2013 )

Abstract    in this paper, free vibration of an Euler-Bernoulli beam with variable cross-section resting on elastic foundation and under axial tensile force is considered. Beam’s constant height and exponentially varying width yields variable cross-section. The problem is handled for three different boundary conditions: clamped-clamped, simply supported-simply supported and clamp-free beams. First, the equation of motion that governs the free vibration is derived and then dimensionless frequencies are determined by using differential transform method (DTM). DTM is a semi-analytical approach based on Taylor expansion series that is powerful tool in solution ordinary and partial differential equations. The effects of axial force, elastic foundation coefficient and non-uniformity parameter on dimensionless frequencies are investigated. Wherever possible, comparisons are made with the studies in open literature. Results show, the DTM yields rapid convergence without any frequency missing although convergence rate depend on boundary conditions. Also, dimensionless frequencies are sensitive to axial force rather than other parameters.


Keywords    Variable Cross-Section Beam, Elastic Foundation, Axial Force, Differential Transform Method


چکیده    در اين مقاله، ارتعاشات آزاد تير اويلر-برنولي داراي سطح مقطع متغير مستقر بر بستر الاستيک و تحت اثر بار کششي محوري مورد بررسي قرار مي گيرد. ارتفاع تير ثابت در نظر گرفته شده و عرض آن به صورت نمايي تغيير مي کند. تحليل براي سه حالات مختلف شرايط مرزي شامل دو سر گيردار، دو سر مفصل و يک سر گيردار-يک سر آزاد انجام شده است. بعد از آنکه معادله ارتعاشات آزاد تير مورد بررسي استخراج شد، فرکانس هاي بدون بعد با استفاده از روش تبديل ديفرانسيلي به دست مي آيند. روش تبديل ديفرانسيلي يک روش نيمه تحليلي مبتني بر بسط سري تيلور مي باشد که به راحتي مي تواند براي حل انواع معادلات ديفرانسيل معمولي و مشتقات جزئي به کار رود. تاثير نيروي محوري، ضريب بستر الاستيک و پارامتر متناظر با تغييرات سطح مقطع بر فرکانس هاي بدون بعد بررسي شده است و در مواردي که در ساير منابع تحليل مشابهي صورت گرفته است، مقايسه بين نتايج انجام شده است. نتايج نشان مي دهد روش تبديل ديفرانسيلي داراي همگرايي سريع بوده و قادر به محاسبه تمام فرکانس ها مي باشد اگرچه سرعت همگرايي آن از شرايط تکيه گاهي اثر مي پذيرد. همچنين فرکانس هاي بدون بعد حساسيت بيشتري نسبت به نيروي محوري از خود نشان مي دهند.


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