ارتعاش آزاد تيرهاي مدرج هدفمند بر بستر ارتجاعي دو پارامتري با استفاده از روش ديفرانسيل كوادريچر

FREE VIBRATION OF FUNCTIONALLY GRADED BEAMS ON TWO PARAMETERS ELASTIC FOUNDATION USING DIFFERENTIAL QUADRATURE METHOD

تيرهاي قرار گرفته بر بسترهاي ارتجاعي، يك مدل پر كاربرد در تحليل و بررسي سازه هاي مهندسي هستند. در دهه هاي اخير، به دليل داشتن مقاومت بالا در برابر حرارت، جذب انرژي و وزن سبك، مطالعاتي در رابطه با مواد نوين همچون مواد مدرج هدفمند انجام شده است. خواص مكانيكي مواد مدرج هدفمند به صورت پيوسته از يك سطح به سطح ديگر و از فلز به سراميك تغيير مي كند. در نوشتار حاضر، براي اولين بار معادلات حاكم بر ارتعاش آزاد تيرهاي مدرج هدفمند بر بستر ارتجاعي دو پارامتري شامل فنر خطي و لايه ي برشي با استفاده از نظريه ي برشي مرتبه ي اول به دست آمده و با به كارگيري روش ديفرانسيل كوادريچر تجزيه شده اند. با تجزيه ي شرايط مرزي با روش ديفرانسيل كوادريچر، اثر شرايط مذكور در معادلات حاكم تجزيه شده بررسي و دستگاه معادلات به دست آمده به حوزه ي بسامد منتقل و بسامد مسيله تعيين شده است. تاثيرير شرايط مرزي، پارامترهاي بستر ارتجاعي، نسبت طول به ضخامت، و ضريب كسر حجمي از موارد بررسي شده در نوشتار حاضر است.

Todays, modern materials are widely used for construction of various structural elements such as beams, plates, shell and the other elements. One of these modern materials is functionally graded materials (FGMs). The uses of these materials are due to their mechanical properties and thermal conductivity. During the two past decades structural elements made from these materials have received wide applications in aerospace, mechanical and civil engineering. Material properties of the functionally graded materials vary continuously from metal on one surface to ceramic on the other surface. The distribution makes these materials applicable in different fields of engineering, especially in environments subjected to high thermal change. For obtaining the governing equations of the structural elements different theories of elasticity can be used. For solving the problems different analytical, numerical and semi-analytical methods can be applied. Among the numerical methods finite element, finite difference, finite volume, differential quadrature and the other methods may be used for solving the problems. The differential quadrature (DQ) method is an accurate, efficient and robust numerical solver with low computational cost. The DQ method has been used for solving free vibration, dynamic analysis and so on. This method can be used for solving structural elements alone or in conjunction with the other analytical or numerical methods. In this study, the method is used for free vibration analysis of functionally graded beams on two parameter elastic foundation. The elastic foundation has linear and shearing layers. The governing equations are derived based on the first order shear deformation theory (FSDT). The governing equations and the related boundary conditions are discretized using the DQ method. Then by employing method of separation of variables, the obtained equations are transferred from temporal domain to frequency domain and the frequency of the beam is calculated. Applicability, rapid rate of convergence and accuracy of the proposed method are demonstrated via solving some examples. Influences of different parameters such as linear and shearing layers, boundary conditions and height-to-length ratio on frequency of the beams are investigated.