利用克拉德尼圖樣研究不同材質平板的聲學色散關係：鋁、黃銅、無氧銅、不鏽鋼、玻璃、木材、壓克力

Abstract

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.