Timoshenko和Euler懸臂梁在流體環境中共振頻之比較

Abstract

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.