靜態輸出回饋與參數更新迭代方法設計最佳目標反應之被動調諧質量阻尼器

Design Passive Tuned Mass Damper With Optimal Target Response Using Static Output Feedback and Parameter Updating Iterative Method

10.6849/SE.202409_39(3).0006

賴勇安(Yong-An Lai)；張淇閎(Chi-Hung Chang)；洪憲証(Xian-Zheng Hong)

TMD ； 最佳被動控制 ； 最佳勁度與阻尼係數 ； 最佳質量 ； 靜態輸出回饋 ； 風力與地震力 ； tuned mass damper (TMD) ； optimal passive control ； optimal stiffness and damping coefficient ； optimal mass ； static output feedback ； wind and seismic loads

結構工程

39卷3期（2024 / 09 / 01）

121 - 153

繁體中文;英文

本文將被動調諧質量阻尼器（tuned mass damper, TMD）針對最佳化目標反應之參數設計問題，轉換為主動控制理論，靜態輸出回饋（或稱直接輸出回饋）中增益矩陣之最佳化問題，經由主動控制理論中最佳增益矩陣之求解方式，從而可得到目標反應均方最小化時，被動TMD之最佳勁度與阻尼係數或最佳質量。所提之靜態輸出回饋設計方法，不論結構為單自由度或多自由度，為有阻尼結構或無阻尼結構，外力為風力或地震力，皆可適用。且針對不同減振目標，僅需選擇不同之輸出矩陣，組合出對應之權重矩陣求解即可，十分直觀與簡便。如結構為單自由度，經由數值模擬驗證，所求出被動TMD針對最佳化目標反應之設計參數，與隨機振動理論下所得之解析解相等，或與近似解相近，確認所提之設計方法無誤且可行。本文亦以所提之方法，另外求得地震力下，單自由度結構速度或絕對加速度之均方反應最小化，TMD之最佳頻率比與阻尼比，供工程師設計時參考；最後，以五層樓之標稱結構與十層樓之ETABS結構，分別加裝被動TMD進行示範，設計TMD之最佳勁度與阻尼係數，或最佳質量，確認此方法對於多自由度結構系統也同樣適用。

This study proposes a comprehensive passive tuned mass damper (TMD) optimization design method to minimize structural mean square responses. The optimization design problem for passive TMD is reformulated as an optimal control problem, specifically, the optimal gain matrix design problem in static output feedback (or direct output feedback). By solving for the optimal gain matrix, the optimal stiffness and damping coefficients, or optimal mass, of the passive tuned mass damper can be obtained. The proposed method is applicable to both single-degree- of-freedom (SDOF) and multi-degree-of-freedom (MDOF) structures, whether damped or undamped structures, and subjected to wind or seismic loads. Moreover, for different vibration reduction objectives, only different output matrices need to be selected, and the corresponding weighting matrices can be combined for the solving process, making it intuitive and straightforward. In the case of SDOF structures, numerical simulations validate that the optimal design parameters of the passive TMD obtained through this method are identical to the analytical solutions derived from random vibration theory, or closely approach to the approximate solutions, confirming the correctness and feasibility of the proposed design method. Additionally, using the proposed method, the optimal TMD frequency ratio and TMD damping ratio for minimizing the mean square response of velocity or absolute acceleration of SDOF structures under seismic forces are presented, providing reference for engineers in design. Finally, demonstrations are conducted with a passive TMD installed on a five-story MDOF structures and a ten-story ETABS structure, respectively, to design TMD optimal stiffness and damping coefficients, or optimal TMD mass. The results confirm that the proposed method is applicable to MDOF structural systems.