應用有限元素頻域法於室內聲響特性之預測

Prediction of the Indoor Acoustic Characteristics by Finite Element Frequency Domain Analysis

李志徹(Chih-Che Li)

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.2009.00526

摘要

本文應用Biot多孔線性彈性動態方程組於拉普拉斯域推導多孔吸音材料之動態複數勁度再計算其吸音係數，並分析空間聲場之特性，文中也針對聲場統御方程式之推導詳加說明。為探討空間聲場之特性，本文將分析方法歸納為二部份，第一部份應用拉普拉斯轉換後之多孔線性彈性動態方程以二維矩形多孔彈性元素及 Galerkin有限元素法推導並進行有限元素頻域分析(FEFDA)。第二部分應用聲場波動方程式配合二維矩形元素進行聲場有限元素分析(AFFEA)。
FEFDA應用空氣層與泡棉材料參數與邊界設定進行一維及二維含有吸音層聲場之響應分析，並與AFFEA結果進行比較。由同網格數之分析結果可發現AFFEA之模態頻率誤差較FDFEA大，且在含有吸音層之聲場分析中，AFFEA需輸入以FEFDA完成之吸音材料聲響阻抗才能進行分析。此外在高頻時AFFEA必須在高網格數下才有較精確之結果。兩種分析亦同時顯示於室內空間聲場設置吸音材料後因受限吸音材料特性，在低頻時效果不佳，但隨著頻率增高後整體吸音效果顯著提升。本文發展之FEFDA也可應用於模擬阻抗實驗，經分析發現模擬之吸音係數也與一維理論解幾近吻合。

關鍵字

室內聲響 ； 動態勁度 ； 頻域 ； 有限元素分析

並列摘要

In this study, Biot's poroelastic dynamic equations were applied to analyze the dynamic stiffness and the sound absorption coefficient of the sound absorption material and to analyze the indoor acoustics. For comparison purposes, the acoustic wave equation was also applied to evaluate indoor acoustics. Two finite element analyses were conducted. First, the poroelastic dynamic equations were transformed to Laplace domain and the Galerkin finite element approach is applied to derive the stiffness matrix of rectangular porous element for conducting the finite element frequency domain analysis (FEFDA). Secondly, the acoustic wave equation is used derive the stiffness matrix of acoustic rectangular element for performing the acoustic field finite element analysis (AFFEA).
Using foam material properties and suitable boundary conditions, the FEFDA was applied to evaluate the frequency response functions of one-dimensional and two-dimensional indoor acoustics and the results were compared with that obtained by the AFFEA. Based on the same element meshes, the results showed that the errors on modal frequencies of AFFEA was large than that of the FDFEA. Furthermore, the AFFEA needs more meshes for obtaining accurate results in high frequency region. In the indoor acoustics analysis of room with sound absorption wall, the AFFEA needs the input of the acoustic impedance of the sound absorption wall, which was obtained from the FDFEA. Analyses showed that subjected to the performance of the absorption material, the sound absorption effect was not good at low frequency region, but was greatly improved at high frequency region. The FEFDA was also applied to simulate impedance experiment in this study and the results were found in good agreement with that obtained from the one-dimensional predictions.

並列關鍵字

Indoor Acoustics ； Dynamic Stiffness ； Frequency Domain ； Finite Element Analysis

參考文獻

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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.