二維彈性超穎材料板之波傳行為探討

Numerical study of two dimensional elastic metamaterial plate

林紀匡(Chi-Kuang Lin)

指導教授 ： 黃心豪

國立臺灣大學/工學院/工程科學及海洋工程學研究所/碩士(2016年)

https://doi.org/10.6342/NTU201602278

摘要

超穎材料是一種人造的特殊材料，透過幾何結構設計使其產生一些特殊性質，例如等效負質量、等效負楊式模數等，且彈性超穎材料的頻散圖中會出現能隙的頻段，若波傳在能隙頻段中傳遞將產生振幅衰減的現象。本研究將焦點擺在能隙的頻段內，以一維負楊氏模數理論模型為基礎，將其簡化並實際製作出超穎材料樑，經過模擬及實驗證明本研究所提出的超穎材料樑能夠使波傳190 Hz至300 Hz之間產生振幅衰減的現象。

接著將超穎材料樑進一步延伸至超穎材料板，預期在頻散圖中產生能隙頻段，因此文中提出一系列設計、理論計算、數值模擬及分析的方法。首先設計出超穎材料板並進行理論推導，推導出模型的運動方程式及頻散方程式，並透過頻散圖分析其能隙的範圍。接著繪製3D模型匯入有限元素分析軟體Comsol 5.0進行模擬分析，並將理論計算與模擬的結果互相比對。最後透過改變材料參數、幾何參數等方式觀察這些參數對能隙範圍及頻段的影響，並根據現今的工程問題提出此模型應用的建議。

經上述方法，本文成功建立超穎材料板的理論頻散方程式，繪製出立體及平面頻散圖，並與數值模擬互相比較，發現兩者針對能隙頻段的分析非常吻合，且證實本研究所提出的超穎材料板在373Hz至440Hz之間會產生能隙。最後藉由參數研究將各項參數對能隙頻段的影響列於表 5 1，透過參數調整後，此模型可應用於減少滯後流對船艦的影響或降低燃料電池汽車運作所產生的噪音。

關鍵字

超穎材料 ； 平板 ； 局部共振 ； 橫波衰減 ； 振動阻隔

並列摘要

Metamaterials are man-made composite materials, with specialized geometrical designs, metamaterials were aimed to possess several unusual properties such as negative mass density, negative Young’s modulus, etc. It was found that bandgaps can be created by the local resonance effect of elastic metamaterials, and the wave propagation will attenuate when propagate in the bandgap. This thesis will focus on the bandgaps. We refer effective negative Young’s modulus metamaterial model, then the model can be further simplified and extended to metamaterial beam (meta-beam). Through simulations and experiments, it is confirmed that the wave amplitude will be attenuated when it propagate through the meta-beam in the frequencies between 190 Hz to 300 Hz.

Then extend the beam to the plate. The methods for modeling, analysis, and design of metamaterial plate (meta-plate) are presented. First, governing equations and dispersion equations are derived, and the bandgap is created in the dispersion curve by the resonance systems. Second, a method to design a 3D model is proposed, and the dispersion curve of the meta-plate is explicitly confirmed by numerical simulations in Comsol 5.0. It is confirmed that the bandgap of the proposed plate model is between 373 Hz to 440 Hz. Third, the parametric studies were also carried out, and listed in table 5-1. Finally, potential applications of the proposed meta-plate are decreasing the damages caused by after-flow of ship or reduce the noise of fuel cell vehicles.

並列關鍵字

Mechanical Metamaterial ； Plate ； Local Resonance ； Transverse Wave Attenuation ； Vibration Isolation

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