基底運動引發板結構動態反應之理論分析與實驗量測;Theoretical Analysis and Experimental Measurement of the Dynamic Response of Plate Structures Induced by Base Excitation

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Title:	基底運動引發板結構動態反應之理論分析與實驗量測;Theoretical Analysis and Experimental Measurement of the Dynamic Response of Plate Structures Induced by Base Excitation
Authors:	劉育愷;Yu-Kai-Liu
Contributors:	機械工程學系
Keywords:	板殼理論;振動;克拉尼圖;暫態波傳;阻尼比;plate theory;vibration;chladni pattern;wave propagation;damping ratio
Date:	2025-01-22
Issue Date:	2025-04-09 18:31:20 (UTC+8)
Publisher:	國立中央大學
Abstract:	本文透過理論分析、有限元素模擬和實驗量測，探討矩形平板受基底運動所引發的頻率域反應與暫態波傳行為，並且藉由雙向耦合有限元素法(FEM)與離散元素法(DEM)模擬顆粒體受平板振動所產生的動態特性，探討基底加速度對顆粒體形成克拉尼圖(Chladni patterns)之影響。本文首先建構基底激振板結構之理論模型，以疊加法求出共振頻率和模態形狀等振動特徵。在實驗量測上，分別使用鋼珠撞擊和振盪器激振兩種方法激發平板動態行為，將平板暫態應變訊號藉由快速傅立葉轉換 FFT 轉換至頻域獲得共振頻率和頻率響應，平板上顆粒體形成克拉尼圖觀察各頻率下的模態形狀。理論、實驗與模擬對於振動特徵的比較中，顯示共振頻率與模態形狀皆具有良好的對應性，此外模擬結果中的節線也與克拉尼圖實驗中顆粒停留位置相吻合，結果也顯示模態形狀的對稱性和特定模態出現的機制有關。暫態波傳分析結合以上振動分析結果為基礎，使用模態展開法，以模態形狀與時間函數建構平板之暫態波傳理論解。本文理論分析均考慮阻尼效應，將平板暫態應變使用短時傅立葉轉換STFT得到訊號各模態的頻率響應衰減趨勢，計算出各模態阻尼比，輸入於考慮阻尼比效應的暫態波傳理論解中，以獲得平板之暫態理論解;實驗量測使用振盪器激振板結構來激發平板動態行為，量測平板因基底激振所產生的暫態應變訊號，將實驗量測結果與有限元素模擬和理論分析結果比較，在定頻激振實驗中，模擬與理論計算出的暫態應變無論是波形或是量值均與實驗十分吻合，而在掃頻激振實驗中，模擬與理論計算出的暫態應變的波形與實驗中有所差異，這是因為實驗系統的共振頻與理論有些許差異所導致的。最後通過多種基底激振波源，驗證考慮阻尼效應的平板暫態波傳理論解之準確性。最後，本文探討四邊自由平板之頻率域反應與顆粒體受平板振動所產生之動態行為。本文使用有限元素模擬探討四邊自由平板之頻率域反應，並比較不同邊界條件產生克拉尼圖之機制。本文採用雙向耦合離散元素法(DEM)與有限元素法(FEM)，模擬顆粒體在彈性矩形板上的動態行為，探討顆粒體在各種基底加速度下的顆粒聚集情況，將模擬結果與實驗進行比較，驗證此方法之準確性。;This study investigates the frequency response and transient wave propagation behavior of a rectangular plate induced by base excitation through theoretical analysis, finite element simulation, and experimental measurement. The frequency response includes vibration characteristics and displacement field response. Additionally, the dynamic characteristics of granular materials induced by plate vibration are explored using a two-way coupled finite element method (FEM) and discrete element method (DEM) simulation, and the influence of dimensionless acceleration on the formation of Chladni patterns in granular materials is examined. In this study, the theoretical model utilizes the superposition method to obtain the natural frequencies and mode shapes. In experimental measurements, both steel ball impact and shaker excitation are used to excite the plate′s dynamic behavior. The transient strain signals of the plate are obtained through FFT transformation to acquire the resonant frequency and frequency response. Simultaneously, the mode shapes at different frequencies are observed through the Chladni patterns formed by the particles on the plate. Comparisons between theoretical, experimental, and simulation results for vibration characteristics show a good correspondence in both resonant frequencies and mode shapes. Furthermore, the nodal lines in the simulation results coincide with the particle resting positions in the Chladni pattern experiments. The results also indicate that the symmetry of the mode shape is related to the mechanism of specific mode appearance. Based on the vibration analysis results, the transient wave propagation analysis utilizes the modal expansion method to construct a theoretical solution for the transient wave propagation of the plate using mode shapes and time functions. The theoretical analysis in this study considers the damping effect, and the short-time Fourier transform is applied to the transient strain of the plate to obtain the frequency response decay trend of each mode. The damping iii ratios of each mode are calculated and input into the theoretical solution considering the damping ratio effect to obtain the transient theoretical solution of the plate. In the experiment, a shaker is used to excite the plate structure to generate transient strain signals during dynamic vibration. By comparing the experimental measurement results with the finite element simulation and theoretical analysis results, it is found that in the constant-frequency excitation experiment, the simulated and theoretically calculated transient strains agree well with the experimental results in terms of both waveform and magnitude. However, in the swept frequency excitation experiment, there are some differences in the waveforms of the simulated and theoretically calculated transient strains compared to the experiment, which is caused by the slight difference between the resonant frequency of the experimental system and the theoretical value. Finally, the accuracy of the transient wave propagation theoretical solution considering the damping effect is verified by applying different types of wave sources to the base of the structure. Finally, this study explores the frequency response of a free-edge rectangular plate and the dynamic behavior of granular materials induced by plate vibration. The finite element simulation is used to investigate the frequency response of the free-edge rectangular plate, and the mechanisms of Chladni patterns generated under different boundary conditions are compared. In this paper, a two-way coupled discrete element method (DEM) and finite element method (FEM) are used to simulate the dynamic behavior of granular materials on an elastic rectangular plate. The particle aggregation under different dimensionless accelerations is investigated, and the simulation results are compared with the experiments to verify the accuracy of this method
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