局部共振模態於聲子晶體平板內之波傳行為研究

Abstract

本論文以數值方法以及雷射超聲波實驗探討具週期性薄膜的聲子晶體平板之頻溝現象以及共振現象的波傳行為。本文使用有限元素法(finite element method, FEM)結合布拉格(Bloch)理論，建立週期性邊界條件來分析聲子晶體結構之頻散關係與特徵位移場。在數值模擬方面，為了解薄膜對於頻散關係的影響，比較頻散曲線中出現的薄膜共振模態時的頻率與同樣尺寸的束制邊界的圓形薄膜的共振頻率，藉由改變厚度來探討，並發現當薄膜厚度逐漸減少，具週期性薄膜的聲子晶體平板的頻散曲線可以視為束制邊界薄膜的共振頻率與未含薄膜的聲子晶體頻散曲線的疊加。另一方面，從特定模態中，觀察到形成相較於反對稱板波約五分之一到五十分之一波速的慢波速與其生成原因。 在實驗方面，本文使用脈衝雷射在基板上激發寬頻之超聲波，並且以氦氖雷射干涉儀來量取縱向位移，實驗結果顯示實驗與模擬結果相當一致，並在薄膜上發現對應於薄膜膜態的顯著共振，並且，引入小波轉換(wavelet transform, WT)來觀察頻溝內共振訊號的慢波速現象，量測結果與慢波模態的穩態相位點(stationary phase point)之波速吻合。 最後，透過本文的模擬和實驗研究可觀察知，藉由週期性薄膜聲子晶體平板顯著的出平面共振現象與慢波速的特性，預期可以做為聲波延遲(acoustic delay)裝置和用於聲波式感測器之延遲線(delay line)縮減，同時可因其高頻振盪的特性加速觸媒與反應物的結合來加速反應時效。

In this thesis, we demonstrate the existence of complete band gaps and the propagation of locally resonant modes in a phononic plate with periodic membranes both numerically and experimentally. In order to understand the influence of the membrane on the dispersion relations, a series of numerical calculations are conducted with the finite element (FE) method. Numerical simulated results show that the dispersion of a phononic plate with periodic membranes can be considered as a superposition of the dispersion of air/steel phononic plate and the resonant frequencies of circular membrane. Besides, in specific bands of locally resonant modes, we found that the group velocity of the modes can be as low as one order of magnitude smaller than that of the A0 mode in a homogeneous flat steel plate of the same thickness. Furthermore, we discussed the mechanism of the slow velocity of the membrane modes in such a structure. On the experiment side, we use a pulsed laser to generate broadband elastic waves and optical devices to detect wave signals. The experimental results are in good agreements with the numerical prediction. Moreover, the slow velocity inside the phononic band gap is observed by using the wavelet transformation, and the results are corresponding to the velocity of the stationary phase point. Finally, according to the numerical simulations and experimental results in this thesis, the intense out-of-plane resonances and the slow velocity can be applied to reduce the length of the delay line and enhance the absorbing rate of the catalyzer in the acoustic wave sensor.