矩形平板於流固耦合問題的振動特性與暫態波傳之理論分析、數值計算與實驗量測

Abstract

本文探討等向性矩形平板在空氣中與流體中的動態特性，結合理論解析、實驗量測和有限元素模擬，分析平板耦合流體之自由振動特性，以及平板在空氣中和流場中受動態外力加載而產生的暫態波傳行為。 理論解析首先確認疊加法分析各種邊界條件的矩形平板振動特性相較於Rayleigh-Ritz method具有較良好的準確性，尤其是存在自由邊界的矩形板振動問題，因此選擇以疊加法所求出的平板共振頻率和模態形狀做為流固耦合振動分析的推導基礎。 流固耦合振動分析以空氣中平板之模態形狀作為基底函數，將流場中平板面外變形視為空氣中模態形狀的線性疊加，此概念可以運用於分析平板單側局部接觸流體和平板完全浸泡在有限域流場中的振動特性。藉由求解聲波方程式獲得流體速度勢函數用於描述流體壓力和速度，以力平衡機制結合流體壓力和平板面外位移，建構出流固耦合系統的頻響函數後可求出流固耦合系統的共振頻率、模態形狀和流體壓力。實驗量測使用電子班點干涉術AF-ESPI和壓電薄膜感測器PVDF，量測水下平板的共振頻率以及拍攝水下平板的全場模態形狀。理論計算皆符合實驗量測和有限元素模擬結果，而結果顯示平板耦合流體後共振頻率和模態形狀都會出現變化，而且流體的水深、邊界和流體壓縮性都會對於平板振動特性產生影響。 暫態波傳分析運用模態展開法的概念，以模態形狀和時間函數建構平板的暫態位移，此解析解可用於分析平板承受任何形式動態外力所產生的暫態行為，例如位移、速度、加速度和應變等。實驗設計一組PVDF壓電薄膜感測器量測真實鋼珠撞擊的波源歷時，輸入至理論解析和有限元素模擬中，並且和實驗量測結果比較，探討動態外力與觀察點位置對於平板暫態行為之影響。本文亦提出從實驗訊號中獲取真實阻尼比的方法，進一步修正理論解析解，提升理論分析平板暫態行為的適用範圍。此理論解也可運用其物理意義從實驗暫態訊號中反求平板上外力作用的撞擊點位置。 本文最後探討平板耦合流體之暫態波傳行為，理論解析以流固耦合振動分析所得的流場中平板模態形狀以及流體壓力以模態展開法的概念代入平板統域方程式中，藉由指數矩陣求解聯立微分程式，獲得流場中平板受動態外力作用的暫態位移和暫態速度。流場中平板暫態位移和速度的理論計算皆符合有限元素模擬結果，此理論解析方法可以從物理意義層面探討流場水深的變化對於平板暫態波傳行為的影響。

This study investigates the dynamic characteristics of a thin rectangular plate in air and coupled with fluid by theoretical analysis, experimental measurements, and finite element calculation. The dynamic characteristics involve the vibration properties and the transient behaviors of plate. In this study, the resonant frequencies and mode shapes of the rectangular plate is determined by the superposition method. The results are compared with the results obtained by the Rayleigh-Ritz method. It is shown that the superposition method can fully satisfy the governing equation and the free edge boundary conditions, and give even more accurate result than Rayleigh-Ritz method. The mode shapes of plate in air obtained by superposition method can then be used as the fundamental function to construct the mode shapes of the plate coupled with fluid. The behavior of the compressible fluid induced by the deformation of the plate is obtained from a three-dimensional acoustic equation. The frequency response equation is derived from the hydrostatic equilibrium between the fluid and plate. Solving the frequency response equation makes it possible to obtain the vibrational properties of the fluid-plate system, such as resonant frequencies, wet mode shapes, and pressure of the fluid. Two experimental methods were employed to measure the vibration characteristics of the thin plate immersed in water. Polyvinylidene difluoride (PVDF) measures resonant frequencies of the fluid-plate system. Amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) was used to measure clear mode shapes of the plate in the fluid. Comparison of the results from theoretical analysis, finite element method, and experimental measurements confirmed the accuracy of our theoretical analysis. This study employed theoretical analysis and experimental measurements in an exploration of the transient behavior of a cantilever plate subjected to impact loading. Theoretical derivation has established that displacement is a product of the time and space functions (mode shapes). The superposition method was used to obtain the mode shapes and resonant frequencies of free vibrations, while the orthogonality of the mode function was used to solve the time function. The variation of the applied external force with regard to time, i.e. force history, was measured experimentally by attaching PVDF sensors. More importantly, it was taken into consideration in our theoretical analysis to determine the transient responses, including displacement and strain. Our results obtained in the theoretical analysis are highly consistent with experimental measurements. This is a clear demonstration of the effectiveness of pairing theoretical analysis with experimentally measured force histories in the representation of transient behavior of a cantilever plate. Based on the vibrational characteristics of a fluid-plate system, a theoretical method is developed to investigate its transient behavior. The simultaneous equations are constructed by the normal-mode expansion. The transient displacement and velocity of the plate coupled with water can be obtained by solving the simultaneous equations.