應用廣義元素疊加法分析多跨度板結構振動、暫態響應與反算問題

Abstract

本論文針對多跨度板結構之動態問題，分別從自由振動分析、暫態響應分析與瞬時外力反算三個面向進行研究，建立一套兼具理論解析與實驗驗證之完整分析架構。多跨度板廣泛應用於高架道路及鐵道橋梁，具有重複跨距與多條中間線支撐，其動態行為較傳統單一矩形平板更為複雜，亟需有效之解析工具與量測及反算技術，以協助結構設計與健康監測。 首先，本論文以古典薄板理論為基礎，採用Gorman所提出之疊加原理，發展廣義元素疊加法以求取不同古典邊界條件組合下多跨度板之共振頻率與模態振型，並與有限元素分析結果比較。數值與實驗結果顯示，所提方法與有限元素分析之共振頻率差異皆小於0.8%，與AF-ESPI量測結果之差異多小於2%。由於解析解具有可微分之特性，其所得多跨度板應變場於應力集中區域較有限元素結果更為連續，有利於後續暫態響應與反算分析之準確性。 其次，基於自由振動模態解析解，本論文推導在考慮與未考慮阻尼效應下，多跨度板受外力作用時之暫態波傳解析解，建立一套可快速預測結構暫態位移與應變響應之理論模型。其計算時間由有限元素分析約70分鐘縮短至約30秒，顯示計算成本可大幅降低。自由振動實驗方面，結合電子斑點干涉術(AF-ESPI)量測共振頻率與模態振型，並與鋼珠撞擊試驗結果相互印證；於暫態波傳試驗中，利用PVDF壓電薄膜感測器定量量測暫態應變歷時，並以布拉格光纖光柵(FBG)感測器之量測結果加以驗證。綜合理論與量測結果可知，本研究所建立之暫態波傳解析解能有效描述多跨度板中之波傳特性及響應隨時間之衰減行為，並進一步據以提出一套阻尼比識別方法，且經由傳統阻尼比評估方式驗證其可行性。結果顯示，考慮阻尼效應後，暫態響應頻譜主共振峰幅值與實驗量測之差異低於5%。 再者，本論文進一步探討瞬時外力之反算問題。首先，利用暫態應變與位移頻譜中各共振頻率在不同撞擊位置與感測點間之貢獻差異，結合已知模態振型與相關係數場，在波源歷時未知之情況下反算撞擊位置，其判定之撞擊點座標經數值模擬與實驗量測驗證，反算位置與實際位置之絕對誤差均小於2 mm，且以薄板對角線長度進行規範化之無因次誤差均小於1.1%。其次，基於考慮阻尼效應之暫態波傳解析解，建立撞擊力歷時與量測暫態應變之線性關係，並採用截斷奇異值分解(TSVD)改善反算運算矩陣之病態性，提出一套可抑制量測雜訊影響之波源歷時重建流程。在代表性算例中，所重建之撞擊力峰值與實際量測結果之誤差小於3%。進一步結合前述撞擊位置反算結果，在撞擊位置與波源歷時皆未知之情境下，仍能有效重建瞬時外力之時域特性。 綜合而言，本論文以理論解析、試驗量測與反算分析，建立一套適用於多跨度板結構之動力分析與外力識別架構，不僅可作為多跨度板設計與性能評估之量化參考，亦可應用於衝擊事件判識與結構健康監測，具備良好工程實務應用潛力。

This dissertation examines the dynamic behavior of multi-span plates through three main components: free vibration analysis, transient response analysis, and the inverse identification of impact forces. Multi-span plates are commonly used in elevated highways and railway bridges, displaying more complex dynamics due to their repeated spans and internal line supports, in contrast to conventional rectangular plates. To support structural design and health monitoring, it is essential to combine analytical methods, numerical simulations, and experimental measurements into a cohesive framework. This study develops a generalized superposition segment method based on classical thin-plate theory and Gorman’s superposition concept to determine the natural frequencies and mode shapes of multi-span plates under various boundary conditions. Numerical and experimental results show that the differences in resonant frequencies between the proposed method and the finite element analysis are all within 0.8%, and the discrepancies from the AF-ESPI measurements are mostly within 2%. It produces smooth, differentiable strain fields that are particularly effective near stress concentrations. Building on the free vibration solutions, we derive transient wave-propagation solutions with and without structural damping, enabling quick predictions of transient displacement and strain responses. Compared to FEM software, the computation time decreases from approximately 70 minutes to roughly 30 seconds, significantly reducing computational cost. Amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) is experimentally utilized to measure resonant frequencies and mode shapes, which are validated through steel-ball impact tests. For capturing transient wave propagation, PVDF piezoelectric film sensors are employed to record transient strain responses. The accuracy of these results is further confirmed using fiber Bragg grating (FBG) sensors. The agreement between theoretical predictions and measurements indicates that the transient solutions effectively describe wave characteristics and temporal attenuation, which supports the identification of damping ratios. The main peak in the frequency-response spectrum predicted by the proposed method differs from the experimental measurements by less than 5%. In addition, inverse methods are developed to reconstruct the impact forces. The impact locations are identified by analyzing the differences in modal contributions within the transient frequency spectra of various impact–sensor pairs, in combination with the known mode shapes and correlation fields. Numerical and experimental results show that the absolute error between the identified and true impact locations is less than 2 mm, and the corresponding dimensionless error normalized by the plate diagonal length is less than 1.1%. The impact force is then reconstructed by relating the unknown load to the measured transient strains, while stabilizing the inverse problem using truncated singular value decomposition. In representative case studies, the error in the reconstructed impact-force peak is within 3% of the experimentally measured value. By integrating location identification with impact force reconstruction, this framework is capable of recovering the time-domain characteristics of impact loads, even when both the impact location and time history are unknown. This demonstrates significant potential for diagnosing impact events and monitoring structural health.