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研究生： 麥凱荻
Madine, Katie Hannah
論文名稱： 獨立彈性結構陣列中的波傳導現象
Wave Propagation in Flexural Systems
指導教授： 嚴大任
Yen, Tajen
Colquitt, Daniel
Colquitt, Daniel
Movchan, Alexander
Movchan, Alexander
口試委員: Haslinger, Stewart
Nieves, Michael
Sharkey, Kieran
張禎元
Morini, Lorenzo
Chang, James
學位類別： 博士
Doctor
系所名稱： 教務處 - 跨院國際博士班學位學程
International Intercollegiate PhD Program
論文出版年： 2023
畢業學年度： 112
語文別： 英文
論文頁數： 225
中文關鍵詞： 波傳導現象 、陀螺 、鋼樑 、負折射
外文關鍵詞： Wave Propagation, Gyroscopes, Beams, Negative Refraction
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    • 本論文對由歐拉-伯努利梁組成的撓性系統的動態行為進行深入研究。 特別關注垂直梁之間的撓性和扭轉旋轉的耦合，以及利用系統特性控制所研究陣列的色散性質的能力。 不對稱性是本論文每項研究中的重複主題，從由應用力選擇引起的晶格上非對稱波形，到後面章節研究的垂直陀螺儀模型的固有手徵性和破碎對稱性。 對於每個系統，分析研究由數值計算和有限元模擬補充，以說明產生的不尋常和反直覺效果。

    論文的前半部研究歐拉-伯努利樑的一維和二維撓性晶格。 研究的標準對象是格林函數，可以從中獲得有關點力和力矩點載荷下晶格動態響應的信息。 在一維和二維晶格中，應用點力和力矩的組合被證明能產生強烈的動態各向異性，以及多種對稱、反對稱、非對稱和單軸波。 特別考慮了歐拉-伯努利梁方格中撓性和扭轉波之間的相互作用； 透過仔細控制晶格元素的慣性和彈性特性，我們證明可以調整晶格的色散特性。 透過實現對波模式的優選方向的控制，我們研究了波在兩個不同的梁晶格之間的界面上的反射和折射。 界面被用來引起負折射、單向反射、光束分裂和模式捕捉內部包含體。

    論文的後續章節研究了由兩個垂直梁組成的垂直陀螺儀模型——垂直梁固定在底座上或安裝在撓性板上——陀螺儀放置在水平樑的自由端。 建立分析模型需要仔細考慮梁接頭之間撓性和扭轉旋轉的耦合，以及陀螺儀旋轉引起的對稱性破壞效應。 我們探討了單一垂直陀螺儀與撓性板上的無限陣列相比的不對稱性，以及由其引起的不對稱性。 我們研究了透過改變系統參數（例如陀螺儀的旋轉速率、光束的長度和陀螺儀軸的方向）來控制陣列的特徵頻率和色散特性。

    這項工作可用於開發旨在透過撓性結構控制波傳播的彈性超材料。 由於彈性波傳播在建築物、橋樑、地震防護裝置、風電場、隱形裝置等物理系統中的廣泛應用，彈性波傳播的研究特別令人感興趣。

    This thesis presents an in-depth study of the dynamic behaviour of flexural systems composed of Euler--Bernoulli beams. Particular attention is paid to the coupling of flexural and torsional rotations between perpendicular beams and the ability to control the dispersive properties of the arrays studied using the properties of the system. Asymmetry is a recurring theme through each study in this thesis, from the asymmetric wave forms on lattices that can be induced by the choice of applied forcing to the inherent chirality and broken symmetries of the perpendicular gyroscope model studied in later chapters. For each system, analytical studies are complemented by numerical computations and finite element simulations, to illustrate the unusual and counterintuitive effects produced.

    The first half of the thesis investigates 1D and 2D flexural lattices of Euler--Bernoulli beams. The canonical object of study is the Green's function, from which information regarding the dynamic response of the lattice under point loading by forces and moments can be obtained. In both the 1D and 2D lattices, the combination of applied point forces and moments is shown to produce strong dynamic anisotropy with a variety of symmetric, anti-symmetric, asymmetric and uni-axial waves. Special consideration is devoted to the interaction between flexural and torsional waves in a square lattice of Euler--Bernoulli beams; by carefully controlling the inertial and elastic properties of the lattice elements, we demonstrate that it is possible to tune the dispersive properties of the lattice. Implementing the resulting control over the preferential directions of wave modes, we investigate the reflection and refraction of waves across interfaces between two dissimilar lattices of beams. The interfaces are used to induce negative refraction, unidirectional reflection, beam splitting and mode trapping inside inclusions.

    Later chapters in this thesis study the perpendicular gyroscope model which is formed of two perpendicular beams---with the vertical beam either clamped at the base or mounted on a flexural plate---and a gyroscope placed at the free end of the horizontal beam. Building the analytical model requires careful consideration of the coupling of flexural and torsional rotations between beam junctions and the symmetry-breaking effects induced by the spinning of the gyroscope. We explore the asymmetry of, and induced by, individual perpendicular gyroscopes compared to infinite arrays on flexural plates. We investigate the control over the eigenfrequencies and dispersive properties of the array afforded by altering parameters of the system, such as the rate of spin of the gyroscope, the length of the beams and the orientation of the gyroscope axes.

    This work has applications in the development of elastic metamaterials designed to control wave propagation through flexural structures. The study of elastic wave propagation is of particular interest due to the wide variety of applications to physical systems such as buildings, bridges, seismic protection devices, wind farms, cloaking devices and more.

    Abstract (Chinese) . . . . . . . . . . . . . . . . . . . . . . . . . . . I Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . II Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . IV Publications and Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . VI Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . VII List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . IX List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . XVII 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1 Lattices and arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.1 Displacement definitions . . . . . . . . . . . . . . . . . . . . 22 2.1.2 Direct and reciprocal lattices . . . . . . . . . . . . . . . . . . 23 2.2 Beam theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.1 Flexural deformation angles in beams . . . . . . . . . . . . . 27 2.2.2 Compressional deformations . . . . . . . . . . . . . . . . . . 28 2.2.3 Flexural deformations . . . . . . . . . . . . . . . . . . . . . 29 2.2.4 Torsional deformations . . . . . . . . . . . . . . . . . . . . . 32 2.3 Kirchhoff–Love plate theory . . . . . . . . . . . . . . . . . . . . . . 34 2.4 Connecting beams to plates . . . . . . . . . . . . . . . . . . . . . . 36 2.4.1 A circular plate with a central inclusion . . . . . . . . . . . 37 2.4.2 Logarithmic boundary conditions . . . . . . . . . . . . . . . 42 2.5 Modelling gyroscopic resonators . . . . . . . . . . . . . . . . . . . . 43 2.6 Generalised Green’s functions on infinite lattices . . . . . . . . . . . 48 3 An infinite chain of Euler–Bernoulli beams 52 3.1 The dispersion equation . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 The Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2.1 The band gap Green’s function . . . . . . . . . . . . . . . . 68 3.2.2 The semi-infinite band gap and the pass band . . . . . . . . 71 3.2.3 Rotational displacement . . . . . . . . . . . . . . . . . . . . 73 3.3 Numerical evaluation and finite element simulations . . . . . . . . . 74 3.3.1 Asymmetric wave modes . . . . . . . . . . . . . . . . . . . . 78 4 A square lattice of Euler–Bernoulli beams 82 4.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2 The dispersion equation . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3 Constructing the band gap Green’s function . . . . . . . . . . . . . 105 4.3.1 Anisotropic behaviour of localised modes . . . . . . . . . . . 109 5 Reflection and refraction at interfaces 116 5.1 Dynamic anisotropy on a homogeneous square lattice of beams . . . 117 5.1.1 Single translational forces and point moments . . . . . . . . 119 5.1.2 Combined applied forces and moments . . . . . . . . . . . . 122 5.1.3 Off-centred asymmetric waves . . . . . . . . . . . . . . . . . 124 5.2 Refraction and reflection at interfaces . . . . . . . . . . . . . . . . . 126 5.2.1 Negative refraction . . . . . . . . . . . . . . . . . . . . . . . 128 5.2.2 Beam splitting . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.2.3 Inclusions and mode trapping . . . . . . . . . . . . . . . . . 134 6 The perpendicular gyroscope 138 6.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.2 Coupling conditions at the junction between the two beams . . . . . 147 6.2.1 The kinematic junction conditions . . . . . . . . . . . . . . . 149 6.2.2 Balancing the forces and moments and the junction point . . 151 6.3 The chiral-torsional boundary conditions . . . . . . . . . . . . . . . 152 6.4 Eigenfrequencies as functions of the gyricity . . . . . . . . . . . . . 156 6.5 Extended chiral-torsional boundary conditions . . . . . . . . . . . . 163 6.5.1 Applying the extended chiral-torsional boundary conditions 167 7 Periodic arrays of perpendicular gyroscopes 170 7.1 Eigenmodes of the perpendicular gyroscope . . . . . . . . . . . . . . 171 7.1.1 Mode shapes of the perpendicular gyroscope . . . . . . . . . 172 7.1.2 Mode shapes for a perpendicular gyroscope on a flexural plate176 7.1.3 Changing the shape of the Kirchhoff plate and orientation of the horizontal beam . . . . . . . . . . . . . . . . . . . . . 178 7.2 Changing the length of the horizontal beam . . . . . . . . . . . . . 183 7.2.1 Similarities and differences between parallel and perpendicular gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . 185 7.3 An infinite array of perpendicular gyroscopes . . . . . . . . . . . . . 188 7.3.1 Dispersion surfaces with increasing gyricity . . . . . . . . . . 190 7.3.2 Asymmetry for perpendicular and parallel gyroscopes . . . . 195 7.3.3 Asymmetry for different lengths of the horizontal beams . . 199 7.3.4 A periodic array of rotated gyroscopes . . . . . . . . . . . . 208 8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

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