حل دقيق ارتعاش آزاد صفحات ايزوتروپ مستطيلي ضخيم بر روي تكيه‌گاه‌هاي ساده با استفاده از توابع پتانسيل تغييرمكان

An Exact Free Vibration Solution of Isotropic Simply Supported Thick Rectangular Plates using Displacement Potential Function

در اين نوشتار، با استفاده از توابع پتانسيل تغييرمكان، حل دقيق ارتعاش آزاد صفحات مستطيلي ايزوتروپ همگن با ضخامت دلخواه ولي ثابت بر روي چهار لبه‌ي ساده ارايه شده و تابع پتانسيل تغييرمكان و به كمك آن معادله‌ي مشخصه‌ي ارتعاش آزاد صفحه با حل دو معادله‌ي ديفرانسيل پاره ‌يي حاكم از مرتبه‌ي 4 و 2 به روش جداسازي متغيرها و اعمال شرايط مرزي به‌دست آمده است. ويژگي عمده‌ي اين پژوهش، عدم وجود محدوديت ضخامت و اعتبار آن براي صفحات نازك، ضخيم نسبي و ضخيم است. به منظور اعتبارسنجي، نتايج حاصل از اين پژوهش با نتايج به‌دست‌آمده از ساير كارهاي تحليلي، كه عمدتاً مربوط به صفحات ضخيم نسبي متكي بر تيوري‌هاي تغييرشكل برشي مرتبه‌ي اول و بالاتر بوده است، مقايسه شده است. نتايج نشان مي‌دهد كه با افزايش ضخامت بسامد بدون بعد كاهش مي يابد كه مقدار اين كاهش در مودهاي بالاتر ارتعاشي با شدت بيشتري همراه است.

In this paper, using displacement potential functions, the exact solution of a three dimension elasticity problem is presented for free vibration of rectangular isotropic plates. It is assumed that the materials of the plate are homogeneous, linearly elastic, arbitrary, but with a constant thickness, and all four edges of the plate are on simple supports. The governing equations in terms of displacement potential functions are two differential equations of fourth and second orders. Assuming harmonic motion and using a separation of variables, the solution of the governing differential equations for displacement potential functions results in exponential and trigonometric expressions along the plate thickness and the other two lengths, respectively. The boundary condition are zero vertical displacement and zero bending moments on all four edges, and all components of stresses, including normal and shear stresses on the top and bottom of the plates, are zero. Applying these boundary conditions result in a characteristic equation of the free vibration of the plate, with which solution, the plate frequency vibration can be calculated. In order to verify the solutions, the obtained results are compared with other analytical work that are largely based on first and higher order deformations theories for moderately thick plates. The most important characteristic of the method presented in this paper is that there is no limitation for thickness in determining free vibration frequency and its validation for thin, moderately thick and thick plates. The investigations have been done for a wide range of aspect ratio (length to width) and thickness to length. The obtained results show that increasing plate thickness decreases nondimensional frequency, and that this decrease intensifies at higher modes of vibration. In addition, investigations show that the Poisson’s ratio has little effect on nondimensional frequency increasing the thickness of the plate. This phenomenon is negligible when the thickness of the plate is decreased.