具額外阻尼之彈性板的導波波傳

Abstract

本研究是研究彈性平板的導波波傳為主，考慮平板本身的熱彈耦合效應，或者是其表面黏滯液體負載所造成的額外阻尼，探討波傳遞時的頻散與衰減關係及其能量耗散。前者代表一個動力系統含有狀態變數對時間t一次微分的阻尼項，後者即為隱含於系統本身彈性係數中的阻尼項。因此，本研究會區分成兩大主題。 第一個主題是探討一受單軸拉伸殘留應力之等向性平板的熱彈導波波傳問題。應用自然、初始與最終三個狀態的聲彈理論，配合傳統熱彈理論，推導在初始狀態下描述的熱彈統御方程式及其導波特徵方程式。利用曲線追蹤法針對複波數的虛部作複數尋根，獲得熱彈導波的頻散與衰減頻譜。由數值結果可發現，除 模態外，其餘模態在某些特定頻率下會出現最小衰減的驟降現象，稱之為Lamé模態，它代表會在厚度方向形成共振，而且在導波傳遞時的能量損耗最小。此外，本文亦探討波傳方向與單軸施力方向的夾角為0°、90°與45°之熱彈導波的頻散與衰減頻譜。 第二個主題是探討表面黏滯液體負載之玻璃平板的導波波傳問題。將黏滯性液體層視為一具剪力剛性c55=-iωη的假想性等向性固體，η為動態黏滯係數，ω為角頻率。經由平板與液體層所架構的雙層結構全域矩陣，利用複數尋根方式獲得頻散與衰減曲線關係，探討平板表面質點運動以及液體層內壓力變化的頻譜特性。除 模態外，其餘模態在頻率接近Lamé模態時會出現最小衰減的驟降現象，在固液界面上的質點軌跡會出現逆轉現象。隨著頻率遞增，平板 模態之位移與應力變化逐漸集中於固液界面，而 模態則是集中於平板下表面。再者，液體層上下表面間壓力差變化及其均勻特性會與平板上表面質點軌跡運動的偏振狀態有直接關係。此外，亦探討液體層厚度改變對相速度頻散與衰減曲線的影響。

This dissertation mainly investigates the dispersion, attenuation, and energy dissipation of ultrasonic guided waves propagating in an elastic plate with excessive damping. The excessive attenuation caused by the thermoelastic coupling of plate or the external viscous fluid loading on the top surface of plate is taken into account. The former represents the damping resulted from the time derivative of state variables in a dynamic system, but the later denotes the intrinsic damping term in the elastic constants. Owing to the above two different excessive damping, the investigation is divided into two works. Thermoelastic waves propagating in an isotropic thin plate exerted by a uniaxial tensile stress are represented in the first work. Characteristic equation of thermoelastic guided waves is formulated based on the theory of acoustoelasticity and classical thermoelasticity. Curve tracing method for complex root-finding is used to determine the attenuation, which is the imaginary part of the complex-value wavenumber. It is found that each plate mode of thermoelastic wave propagating in an isotropic plate with or without pre-stress has a minimum attenuation at a specific frequency except the A0 mode. These modes are called by the Lamé modes, which are the volume resonances in the thickness direction and propagate along the plate with the least energy dissipation. Frequency spectra of the phase velocity dispersion and attenuation of thermoelastic waves propagating along various orientations in the uniaxial pre-stressed thin plate have further been discussed. The second work describes an investigation of acoustic guided wave propagation in a glass plate overlain with a poly-vinyl-alcohol (PVA) layer. The PVA layer is modeled as a hypothetical isotropic solid with dynamic viscosity. Dispersion and attenuation curves, mode shape, trajectories of surface particles on the substrate, and pressure in the fluid layer are studied numerically. Except for the A0 mode, a steeply decreasing attenuation and a reverse trajectory of motion are observed near the frequency of Lamé mode for the different modes. With increasing frequency, displacement, stress, and energy of the A0 mode are significantly confined to a region near the top surface of the plate. A similar phenomenon occurs near the bottom surface for the S0 mode. The pressure gradient and its distribution in the fluid layer are directly related to the trajectories of surface particles on the interface of fluid and substrate. The symmetric modes, except for the S0 mode, at frequencies corresponding to the maximum group velocity, are the appropriate choices for generating uniform acoustic pressure in the fluid layer. Moreover, a glass substrate overlain with a glycerin layer is also taken in account, and its frequency spectra of the phase velocity dispersion and attenuation have further been discussed.