Timoshenko和Euler懸臂梁在流體環境中共振頻之比較

指導教授：張正憲臺灣大學：應用力學研究所丁尚洋Ting, Shang-YangShang-YangTing2014-11-302018-06-292014-11-302018-06-292014http://ntur.lib.ntu.edu.tw//handle/246246/264097本文主要探討在黏滯流體環境中 Timoshenko梁的振動模型。並與Euler梁在黏滯流體環境中的振動行為分別以結構尺寸、流體環境、模態階數、剪切模數作比較。首先介紹相關文獻，利用任意截面不可壓縮黏滯流體理論，計算流體施加在扁平梁的水力負載，再將水力函數耦合進Timoshenko梁理論中，取得流固耦合後頻率響應的求解。同時也介紹Euler梁理論流固耦合後的頻率響應。藉由物理行為以及文獻中的結果與兩理論之間的關係相互驗證，以數值結果分別比較兩理論於不同的梁結構尺寸、不同模態階數、不同流體黏滯性、不同流體密度、不同的剪切模數之關係。以微懸臂梁作為感測器在流體環境作量測時，Timoshenko梁理論與Euler梁理論的差異會比真空中的更大，另外為了有較好的量測靈敏度需要在高模態階數下操作或是需要使用更粗短的懸臂梁時等這些情下時，使用考慮了剪切變形與轉動慣量的Timoshenko梁理論來作計算會比Euler梁理論更為妥適。This paper build the vibration model of Timoshenko beam in viscous fluid. To compare the behavior with the vibration model of Euler beam in viscous fluid by the size, order of modes, fluid environment and shear modulus. Using the Green''s function to solve incompressible viscous fluid theorem of any cross-section, and to conclude hydrodynamic loading that fluid applied to the flat beam. After that, have the hydrodynamic functions coupled with the Timoshenko beam theory to obtain the frequency response function and the frequency response function which affected by the added mass after fluid-structure interaction. The Timoshenko beam numerical results are verified with the physical behavior, the relationship between the two theories and the numerical results of the reference. The numerical results of the frequency response functions were compared the relationship to two theories in different sizes of beam, different density and viscosity of fluid environment, different shear modulus, and the different order of modes. When using the micro-cantilever sensors in the fluid environment for measurement, the difference between Timoshenko beam theory and Euler beam theory will be greater than the vacuum. In addition, if we want to get the better sensitivity, we need to measure in higher mode and use the thicker beam. At such times, using the Timoshenko beam theory which considers the shear deformation and the moment of inertia can make calculations more appropriate than Euler beam theory.目錄摘要 III Abstract IV 圖目錄 VII 表目錄 XII 符號表 XIII 第一章緒論 1 1-1研究動機與目的 1 1-2文獻回顧 1 1-3論文架構 5 第二章黏滯流體水力函數 6 2-1 黏滯流體下任意截面流固耦合 6 2-2 黏滯流體下扁平樑的水力負載 11 第三章流固耦合振動系統與頻率響應 15 3-1 Euler梁 15 3-1-1 結構統御方程式 15 3-1-2 頻率響應及振型 17 3-1-3 流固耦合後頻率響應函數 20 3-2 Timoshenko梁 23 3-2-1 結構統御方程式 23 3-2-2 流固耦合後的共振頻率 25 3-2-3 Timoshenko梁的振型 31 第四章理論驗證與數值結果 33 4-1 理論驗證 33 4-1-1 靜態分析 33 4-1-2 Timoshenko梁與Euler梁在真空環境中的比較 36 4-1-3 與文獻比較 39 4-2 數值結果 43 4-2-1流體對於振動的影響 43 4-2-2結構長度厚度比值及模態階數與流體對兩理論差異的影響 48 4-2-3流體黏滯性對兩理論差異的影響 57 4-2-4流體密度對兩理論差異的影響 66 4-2-5剪切模數對結構與兩理論差異的影響 75 第五章結論及未來展望 84 5-1結論 84 5-2未來展望 86 參考文獻 87 附錄 911633106 bytesapplication/pdf論文公開時間：2019/02/26論文使用權限：同意有償授權(權利金給回饋本人)微懸臂梁感測器原子力顯微鏡振動頻率響應Timoshenko梁理論Euler梁理論黏滯流體Timoshenko和Euler懸臂梁在流體環境中共振頻之比較Comparisons of Resonant Frequency between Timoshenko and Euler Cantilever Beam Immersed in the Fluid Environmentsthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/264097/1/ntu-103-R00543044-1.pdf