應用遞迴最小平方差法識別時變系統之瞬時模態參數

Abstract

本研究為利用線上(on-line)分析之數種遞迴最小平方差法(RLS)從時變系統之動態位移反應架構TVARX(time varying autoregressive with exogenous input)模型，利用識別之時變係數直接估算瞬時模態參數。所考慮之遞迴最小平方差法有遺忘因子法、卡式濾波器，以及權重多項式基底函數法。 藉由單自由度之時變線性系統之數值模擬反應驗證各種遞迴最小平方差法之優劣點，其中探討遺忘因子(λ)、卡式濾波器之控制因子( )、多項式基底函數之項數(k)、TVARX之模型階數(I,J)與雜訊等參數對識別結果之影響。分析結果顯示 (1)權重多項式基底函數法中的遺忘因子(λ)之選擇對於瞬時模態的識別結果相當敏感，(2) 當分析2% 噪訊比之反應時，所使用各種遞迴法之識別效果不盡理想，但是仍可合理追蹤瞬時模態之變化趨勢。 最後分別應用卡式濾波器以及遺忘因子法於分析鋼筋混凝土門型架構和三層樓鋼構之震動台實測資料，識別其瞬時模態特性；識別結果與觀察到之物理現象一致。

This study applies various recursive least squares (RLS) algorithms to establish the time varying autoregressive with exogenous input (TVARX) model for a structure from its dynamic displacement responses and further estimates the instantaneous modal parameters of the structure from the time varying coefficients of TVARX model. The RLS algorithms under study include a forgetting factor approach, a Kalman filter approach and a weighted polynomial basis function expansion approach. The RLS algorithms under consideration are utilized to process numerically simulated responses of single-degree-of-freedom (SDOF) systems with different instantaneous natural frequencies and modal damping ratios. The advantages and shortcomings of these RLS algorithms are investigated through comprehensively studying the effects of main parameters in each approach on accurately identifying the instantaneous modal parameters of the SDOF systems. The parameters under investigation include the order of TVARX model, forgetting factor, control factor in Kalman filter, order of polynomial basis, and noise. It is found that the identified results obtained by the weighted polynomial basis function expansion approach are very sensitive to the values of forgetting factor. Although the instantaneous parameters are poorly identified by all the approaches considered herein in processing the noisy responses with 2% noise-to-signal ratio, the general trends of instantaneous modal parameters varying with time are reasonably found. The approaches under consideration are further applied to process measured dynamic displacement responses of a single-story RC frame and a three-story steel frame in shaking table tests. The specimens were damaged during testing. The identified trends of instantaneous modal parameters varying with time are consistent with the observed physical phenomena.