矩形水槽底板自由振動之研究

A Study of Free Vibration of a Rectangular Bottom Plate of a Water Tank

本文研究有關矩形水槽底板（rectangular bottom plate of a tank）之自由振動問題，依古典板理論（Kirchhof theory）探討矩形水槽底板在空氣中之自由振動，使用Rayleigh-Ritz近似法，計算矩形板之自然頻率。對矩形水槽底板之流體結構耦合（fluid-structure coupled）效應，是以流體在無黏性、不可壓縮、無旋性及無自由液面條件下，依勢流理論，導出速度勢的積分方程式，應用傅立葉級數（Fourier series）展開，得到速度勢（velocity potential）函數，再以能量法計算板與流體耦合之自然頻率，將理論值與相關文獻的理論值比對，更進一步用實驗驗證。結果顯示出矩形板在水中自然頻率為空氣中百分之三十到百分之七十之間。從本文得到的理論公式，將以不同邊界條件之矩形板，計算出在空氣中及水中之無因次頻率參數，在工程設計時，可易於得到矩形板在空氣中及水中的自然頻率。 This paper studied the free vibration of the rectangular bottom plate of a water tank. The Kirchhof plate theory was used for the free vibration of the rectangular plate in the air. The natural frequency of the free vibration of the rectangular plate was calculated by the Rayleigh-Ritz approximation method. The fluid is assume to be invisicid. incompressible, irrotational, and the effect of surface waves was neglected. Effect of fluid-structure coupling at the rectangular bottom plate of the water tank was modeled by potential-flow theory, and with Fourier series expansion, the integral equation can be derived, and then the velocity potential can be obtained. The energy method was then used to calculate the natural frequency with effect of the fluid-plate coupling. The theoretical non-dimensional frequency parameters were compared with those from literatures. And tests were conducted to verify the predictions. The results show that the natural frequency of the rectangular bottom plate of the filled water tank is about 30%~70% of that in air. For the convenience of computing nature frequencies, a table of non-dimensional frequency parameters of the free vibration of the rectangular plate in air and in the water has been created based on boundary conditions of the rectangular bottom plate.