臺灣博碩士論文加值系統

本論文期望提出一套方法以簡潔的理論快速且精準分析平板振盪實驗的結果以重建材質的共振模態與色散關係。過去，共振模態的建立需仰賴數值疊代方法逼近實驗量測之共振頻率數據。欲決定該材質的色散關係更需優先量測出關鍵的彈性係數。然而，數值疊代方式往往耗時且計算量龐大，彈性係數的決定也須透過大量實驗數據統計才能得出相對精準的結果。這也使得傳統方式難以至針對個別狀況進行快速分析。在本文方法中，我們透過解析的理論模型計算出各共振頻率下克拉德尼平板振盪實驗的節線圖案。藉由與實驗結果一對一的完美對應以確認共振模態的重建。我們進一步聯結重建共振波數與實驗共振頻率的關係建構出鋁、黃銅、純銅、不鏽鋼、玻璃、木板和壓克力等常見材質的聲學色散關係。

This thesis propose a method to rapidly and accurately reconstruct the resonant modes and dispersion relationships of thin plates in different materials. In the past, the reconstructions of resonant modes are usually fulfilled by utilizing some approximative method based on numerical iteration to match the experimental resonant frequency spectrum. Besides, the measurements of key elastic coefficients of material, e.g. the Young’s modulus and Poisson ratio, are necessary for the determination of acoustic dispersion relationship. However, not only the numerically iterative process requires tedious calculations which takes lots of time, but the precision of elastic coefficients depend on a large amount of statistics on experimental data. As a consequence, rapid analysis of resonant modes and dispersion relationship are hard to achieve case-by-case by the traditional method. In this work, we analytically develop a theoretical model to calculate the Chladni figures of thin plates. We show the experimental resonant modes can be perfectly reconstructed once the theoretical nodal patterns reveal one-to-one correspondence to the experimental observations. We further demonstrate the dispersion relationships of thin plates in different materials such as aluminum, brass, copper, stainless steel, glass, wood and PMMA can be easily determined by linking the resonant frequencies to the reconstructed wavenumbers.

中文摘要....................................... i
英文摘要....................................... ii
誌謝.......................................iii
目錄.......................................iv
圖表目錄.......................................vi

一、 緒論……………………………………………………1
1.1 研究動機………………………………………………1
1.2 論文架構………………………………………………5
二、 實驗儀器及方法………………………………………6
2.1 平板振盪的歷史發展…………………………………6
2.2 共振頻譜的量測方法………………………………………11
2.3 共振節線圖樣的量測方法…………………………………14
三、 共振模態的理論基礎………………………………………16
3.1 單體振盪……………………………………………………18
3.2 單體共振……………………………………………………19
3.3 連續體振盪-繩波………………………………………… 23
3.4 二維連續體共振的理論模型………………………………28
四、 平板共振模態的理論重建…………………………………32
4.1 共振模態的重建方法………………………………………32
4.2 平板色散關係的建立………………………………………34
五、 不同材質平板共振模態及色散關係的建立………………38
5.1 常見材質特性………………………………………………38
5.2 不同材質共振頻譜的量測…………………………………44
5.3 不同材質共振節線圖樣的紀錄……………………………47
5.4 不同材質的共振模態及色散關係………………………50
5.5 反相疊加節線圖樣的理論修正………………………55
六、 結論與未來展望……………………………………………63
6.1 結論…………………………………………………………63
6.2 未來展望……………………………………………………64

參考文獻 65

附錄一 儀器本身的頻率與電流變化……………………………69
附錄二 鋁平板的頻率與電流變化…………………………………79
附錄三 青銅的實驗共振圖樣………………………………………89
附錄四 黃銅的實驗共振圖樣………………………………………91
附錄五 無氧銅的實驗共振圖樣……………………………………93
附錄六 不鏽鋼的實驗共振圖樣……………………………………95
附錄七 玻璃的實驗共振圖樣………………………………………97
附錄八 壓克力的實驗共振圖樣……………………………………97
附錄九 木板的實驗共振圖樣………………………………………97
附錄十 青銅的圖樣比對……………………………………………98
附錄十一黃銅的圖樣比對……………………………………………99
附錄十二無氧銅的圖樣比對…………………………………………100
附錄十三不鏽鋼的圖樣比對…………………………………………101
附錄十四玻璃的圖樣比對……………………………………………102
附錄十五壓克力的圖樣比對…………………………………………103
附錄十六木板的圖樣比對……………………………………………103
附錄十七青銅反相節線圖樣比對……………………………………104
附錄十八黃銅反相節線圖樣比對……………………………………105
附錄十九無氧銅反相節線圖樣比對…………………………………106
附錄二十不鏽鋼反相節線圖樣比對…………………………………107

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