應用有限元素頻域法於含飽和液體多孔薄板之彎曲振動分析

On the Flexural Vibration of Thin porous plate Saturated with Fluid by Finite Element Frequency Domain Analysis

陳央澤(Yang-Ze Chen)

指導教授 ： 蔡慧駿

淡江大學/工學院/機械與機電工程學系碩士班/碩士(2009年)

https://doi.org/10.6846/TKU.2009.00529

摘要

本文使用Biot多孔彈性理論推導多孔薄板之彎曲振動統御方程組與頻域多孔薄板元素剛性矩陣，以探討多孔薄板之彎曲振動行為。
文中應用多孔彈性理論，於平面應力假設下推導多孔薄板之統御方程組，再於拉普拉斯域中推導多孔薄板三角與矩形元素之剛性矩陣，並藉由衝擊負荷作用與各式邊界條件限制完成多孔薄板之有限元素頻域分析。由多孔薄板統御方程組之比較驗證及多孔薄板有限元素頻域分析之模態頻率與模態行為結果與理論值之比較顯示，本研究建立之有限元素頻域分析確可準確模擬多孔薄板受特定及彈性邊界支撐限制之彎曲振動行為。
多孔薄板因內含之液體與固體架構耦合作用而有特殊之動態消散特性。由多孔薄板撓度頻率響應之模態振幅衰減結果顯示消散係數愈大其振幅影響愈顯著，同時增加液體體積模數也顯著提升多孔薄板之模態頻率。因此藉由飽和液體之改變將可調整多孔薄板之模態頻率與振幅，進而達到振動控制之目的。最後研究經無因次分析了解無因次參數變異於多孔薄板第一模態撓度頻率響應的影響。經分類發現無因次材料參數的影響大致可分為起始振幅、模態振幅及模態頻率三大部份。

關鍵字

多孔薄板 ； 彎曲振動 ； 彈性支撐 ； 頻域 ； 有限元素分析

並列摘要

In this study, Biot's poroelastic theory is used to derive the governing equations of flexural vibration of thin porous plates as well as the stiffness matrixes of the frequency domain thin porous plate elements.
First, the poroelastic theory is used to formulate the governing equations of flexural vibration of thin porous plates based on the plane stress assumptions. Then, the governing equations are transformed to Laplace domain and the Galerkin finite element approach is applied to derive the stiffness matrixes of triangular and rectangular porous elements. After applying impulsive loadings and boundary conditions, the finite element frequency domain analysis of thin porous plates can thus be accomplished. Upon examining the governing equations and the modal frequencies and mode shapes of thin porous plates with specified boundary conditions or elastic restraints, it is validated that the finite element frequency domain analysis can obtain good flexural vibration results for thin porous plates.
A thin fluid-saturated porous plate could present typical dissipation effects owing to the interaction of the fluid and the solid skeleton. Upon examining the reduction of the modal amplitudes after the increase of the fluid’s viscosity, it is learned that the more the increase in dissipation effects the more the reduction in modal amplitudes. It is also found if the bulk modulus of the saturated fluid is increased the plate’s modal frequencies are increased. Accordingly, the modal frequency and the modal amplitude can be adjusted by changing the properties of the saturated fluid, and the vibration control of thin porous plates can thus be achieved. In the end of this study, the influences of dimensionless coefficients on the first mode of a thin porous plate are examined and the results are discussed into three categories signifying the effects on the beginning amplitude, the modal amplitude, and the modal frequency.

並列關鍵字

Thin Porous Plate ； Flexural Vibration ； Elastic Restraint ； Frequency Domain ； Finite Element Analysis

參考文獻

2. K. Terzaghi, “Theoretical Solid Mechanics”, Wiley, New York, 1943.

3. M. A. Biot, “General Theory of Three-Dimensional Consolidation”, Journal of Applied Physics, Vol. 12, pp. 155-164, 1941.

4. M. A. Biot, “Theory of Elasticity and Consolidation for a Porous Anisotropic Solid”, Journal of Applied Physics, Vol. 26, No. 2, pp. 182-185, 1955.

5. M. A. Biot, “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low-Frequency Range”, The Journal of the Acoustical Society of America, Vol. 28, No. 2, pp. 168-178, 1956.

6. M. A. Biot, “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. II. Higher Frequency Range”, The Journal of the Acoustical Society of America, Vol. 28, No. 2, pp. 179-191, 1956.