傳統的亞伯拉罕時域法(ITD)為建構於系統自由響應下的時域唯輸出模態估測方法。前人已利用相關函數技術將ITD法推廣至隨機響應下的系統識別，但往往受限於選定的參考頻道內含模態激發豐富程度而影響模態估測的有效性。吾人利用協方差矩陣內含所有量測頻道間響應數據的相關性，即保有完整的模態信息；再結合奇異值分解法，可藉由奇異譜分析判定系統最小階數，然而可能會因模態激發不良或系統階數選定不佳，導致影響識別可靠性。本研究將協方差矩陣引入ITD法，可避免系統階數選定影響模態參數估測，並針對受具緩慢變化乘積模型非定常之外力激勵於線性系統，以非定常響應建構協方差矩陣，藉由協方差矩陣降低外力與雜訊對於系統響應的影響，使ITD法的適用性延伸至非定常過程。最後，利用數值模擬及實驗驗證本研究所提方法的有效性，甚至對於具相近模態的結構可有效估測系統模態。

The conventional Ibrahim time domain method (ITD) using free response is a response-only modal identification technique. Previous studies have shown the ITD method in conjunction with correlation technique can be applied to the modal estimation from random response data only. However, the identification effectiveness is easily influenced by the choice of the reference channel. In this study, we use the covariance matrix consisting of correlation functions among response data for all measurement channels and avoid the choice of single reference channel in correlation technique. The minimum order of the system can be determined through the singular spectrum analysis. The poor identification may be caused due to the improper system order and incomplete modal information from insufficient frequency content around the structural modes of interest. By introducing the covariance matrix to the procedure of ITD method, the effectiveness of modal estimation can be improved without the estimation of system order. The data correlation matrix is constructed by covariance matrix in state space form composed of the non-stationary response in the form of the product model with the slowly-time-varying function and then extends the ITD method to modal identification directly and solely from nonstationary response data. Numerical simulations and experimental verification of effectiveness of the proposed method for modal identification of structural systems, even with closely spaced modes.

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