沉浸在不同流體之剪切變形梁共振頻率的一階及三階理論研究

Study of resonant frequencies of the first and third order shear deformation beam theories immersed in fluids

陳⿬璋(Muh-Jang Chen)

指導教授 ： 張正憲

國立臺灣大學/工學院/應用力學研究所/碩士(2015年)

https://doi.org/10.6342/NTU.2015.02916

摘要

本文致力於應用不同梁理論建立懸臂梁和橋式梁沉浸於流體中振動的模型，佐以文獻實驗數據，加以比對驗證。首先以古典梁搭配Sader之水力函數建立基礎模型。接著應用一階及三階剪切變形理論搭配相同之水力函數建立模型。應用兩種梁的邊界之後，以掃頻的方式找出剪切變形理論之共振頻。最後長厚比和材料參數對剪切變形理論和古典梁理論之差異造成的影響，並探討在長厚比和材料參數下，三種環境對三種梁理論之共振頻差異的影響。

關鍵字

一階剪切變形理論梁理論 ； 三階剪切變形理論梁理論 ； 微懸臂梁 ； 橋式梁 ； 共振頻

並列摘要

This thesis studies the resonant frequencies of cantilevered and fixed-fixed (bridge) beams immersed in fluids using 1st order and 3rd order shear-deformable beam theories. First, the classic model is developed under Euler-Bernoulli beam theory (EBT) and the hydrodynamic function presented by Sader. Second, the Timoshenko beam theory (TBT) which is a first order shear deformation beam theory and the Reddy beam theory (RBT) which is a third order one are applied to develop new models for biosensors. To obtain the resonant frequencies, boundary conditions of cantilever and bridge beams are both presented. Third, the theoretical prediction developed in this thesis is compared with the experimental data in the literature. Forth, this work is devoted to investigating the effects of aspect ratio and material coefficient ratio on the differences of resonant frequencies obtained from different models. Finally, to investigate the influences in fluids with different viscosities, water and glycerin are considered.

並列關鍵字

Timoshenko beam theory ； Reddy beam theory ； micro-cantilever ； micro-bridge ； resonant frequency

參考文獻

[2] Lochon, F., Dufour, I., & Rebiere, D. (2005). "An alternative solution to improve sensitivity of resonant microcantilever chemical sensors: Comparison between using high-order modes and reducing dimensions," Sensors and Actuators B: Chemical, 108(1), pp. 979-985

[3] Tuck, E. (1969). "Calculation of unsteady flows due to small motions of cylinders in a viscous fluid," Journal of Engineering Mathematics, 3(1), pp. 29-44.

[5] Van Eysden, C.A., & Sader, J.E. (2006). "Small amplitude oscillations of a flexible thin blade in a viscous fluid: Exact analytical solution," Physics of Fluids (1994-present), 18(12), # 123102.

[6] Timoshenko, S.P. (1921). Lxvi. "On the correction for shear of the differential equation for transverse vibrations of prismatic bars," The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 41(245), pp. 744-746.

[8] Levinson, M. (1981). "A new rectangular beam theory," Journal of Sound and vibration, 74(1), pp. 81-87.