矩形平板部分浸泡於流場中的振動特性、暫態波傳與反算問題之理論解析、數值計算與實驗量測

Theoretical Analysis, Numerical Calculation and Experimental Measurement on Vibration Characteristic, Transient Wave Propagation and Inverse Problem of a Rectangular Plate Partially Submerged in a Fluid

陳冠瑋(Guan-Wei Chen)

指導教授 ： 馬劍清

國立臺灣大學/工學院/機械工程學研究所/博士(2021年)

https://doi.org/10.6342/NTU202102835

摘要

本文透過理論分析、實驗量測和有限元素法模擬，探討等向性矩形平板部分浸泡於有限域流場中之自由振動特性，以及平板於空氣中和流場中受動態外力加載而產生的暫態波傳行為，最終將本文提出的解析解運用於反算問題中，提供更廣泛的工程應用價值。在振動分析上，本文使用疊加法求解矩形平板於空氣中面外方向自由振動的共振頻與共振模態，再將空氣中平板模態形狀(乾模態)作為基底函數，透過特徵函數展開法將流場中平板的面外模態形狀(溼模態)以乾模態形狀作線性疊加。在流體分析上，利用分離變數法求解不可壓縮、無黏且無旋流場的流體速度勢函數，並將流體壓力和平板面外位移經由力平衡機制建構出頻響函數，求得水下平板的共振頻率及模態形狀。水下振動實驗部份，本文運用電子斑點干涉術(AF-ESPI)和光纖光柵感測器(FBG)量測水下平板的共振頻與溼模態，藉此觀察水下模態的扭曲情形以及探討有限流場牆邊界對平板振動特性的影響，並將理論分析、實驗量測及有限元素數值模擬的結果進行比較。透過本文所提出的理論解析法，能直觀地了解到每個溼模態係由乾模態所疊加而成，透過其貢獻係數與扭曲參數能進一步探討溼模態與乾模態的關係，藉此研究水下平板溼模態的扭曲情形。暫態分析部分，本文運用模態展開法將平板的模態形狀視為基底函數，搭配空氣中模態的正交特性，藉此分析平板承受動態外力載荷作用下所產生的暫態波傳行為。暫態實驗量測上，本文使用一組PVDF壓電薄膜感測器量測鋼珠撞擊平板所產生的外力波源歷時，再利用光纖位移計(FS)及光纖光柵感測器(FBG)，分別量測水下平板的暫態位移及暫態面內應變，並將實驗量測結果和有限元素法模擬及理論分析結果比較，探討流場水深對於平板暫態行為之影響。接著，本文利用解析解連續可微分的優勢，將所提出的理論解析法運用於各類反算問題上。運用簡單體法，可以透過實驗量測的共振頻反算出水下平板的材料及幾何參數，利用梯度下降法，能從實驗量測之暫態位移反算求得外力衝擊位置。此外，基於本文的水下暫態波傳解，能推導出波源歷時與暫態響應之關係式，透過理論解析方法反算求得平板所承受的動態外力之波源歷時。最終，本文透過理論解析研究流場表面波對流場-平板耦合系之統振動特性的影響，並探討流體與固體在強耦合效應下的振動行為與模態扭曲之情形。總結而言，本研究運用理論解析、實驗量測和有限元法模擬，提供水下平板耦合系統之動態行為的深入研究及探討，並將所提出的理論解析法運用於各類常見反算問題上，提供工業界設計及檢測的相關運用價值。

關鍵字

流固耦合 ； 振動 ； 暫態波傳 ； 特徵函數展開法 ； 模態展開法 ； 電子斑點干涉術 ； 反算問題

並列摘要

The objective of this study was to explore from a theoretical perspective vibration characteristics and transient response of a plate submerged in a fluid, and apply the proposed analytical solution in dealing with the inverse problems. The efficacy of the proposed scheme was evaluated via numerical simulation and experimental measurements involving the vibration analysis and transient wave propagation of a fluid-plate interaction system. In the vibration analysis, the superposition method was exploited to obtain analytical solutions for the free vibration of rectangular plates. Using the eigenfunction expansion method, the underwater displacement of plates (i.e., wet mode shapes) was expressed as expansions of dry mode shapes (i.e., mode shapes in a vacuum). The derivation of wet mode shapes from dry mode shapes proved effective in analyzing problems for plate-fluid interactions. Our analysis revealed that every wet mode shape is composed of multiple dry mode shapes with corresponding contribution coefficients. In terms of the flow field, the dynamic behavior of an inviscid, incompressible, and irrotational fluid was expressed by the velocity potential. The effect of surface waves was also investigated by solving fluid velocity potential associated with bulging and sloshing modes. Underwater experiments were conducted using non-contact, whole-field optical technique (AF-ESPI) to provide excellent fringe patterns from which to observe distortions in the mode shapes. The effect of a solid wall on the vibrating plate was also investigated by the fiber Bragg grating (FBG) sensors. The accuracy of the results obtained using the proposed theoretical approach was verified via comparison with numerical results obtained using the finite element method (FEM) and experimental measurements. In the transient analysis, normal mode method was used to derive the transient response of immersed plates by superposing all resonant modes. The relationship between dry mode shapes and wet mode shapes was essential to the concept of proposed methodology. To quantitatively analyze the transient behavior of the plate, the PVDF film sensors were employed to measure the impact history. The transient displacement and in-plane strain were measured by the fotonic sensor (FS) and polyvinylidene fluoride (PVDF) film sensors, respectively. Moreover, the FBG sensor was employed to measure the underwater plate’s transient strain, which performed a high quality for being an underwater sensor. Comparing to the simulation and experimental results, our proposed method was validated in analyzing the transient behavior of a submerged plate. Finally, proposed analytical solutions were used to deal with several inverse problems, including inverse estimation of material parameters, inverse identification of impact locations, and inverse reconstruction of impact history. Due to the differentiability of analytical solutions, a high accuracy of inverse results was presented. Instead of using optimization method, the impact history could be solved theoretically from the transient displacement of plates. To conclude, this study may have contributed to better understanding of vibration characteristics and transient response of the fluid-plate interaction system, as well as providing theoretical methodologies to deal with vibration analysis, transient wave propagation, and inverse problem.

並列關鍵字

fluid-structure interaction ； vibration ； wave propagation ； eigenfunction expansion ； normal mode expansion ； AF-ESPI ； inverse problem

參考文獻

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