奇值分解法對辨識模態參數之影響分析

An Analysis of SVD on the Identification of Modal Parameters

程安邦(An-Pan Cherng)

《中華民國振動與噪音工程學會論文集》 1997 (1997/05) Pp. 361-372

https://doi.org/10.30028/CSSVANNUAL.199705.0361

摘要

奇值分解法(Singular Value Decomposition, SVD)在數值分析上屬於穩定但費時的矩陣分解方法。SVD將原矩陣分解爲三個矩陣的乘積－前後爲正交矩陣，中問則爲對角矩陣。正交矩陣將不相關的訊號分離，而對角矩陣中的奇值則代表了各訊號的強度。以時域振動反應來擷取模態參數的方法中，即有許多利用了SVD的上述特性。本文分別以理論與模擬例說明這些奇值在適當情形下與位移訊號矩陣內各模組之振幅係數成正比、與頻率及阻尼係數成反比。不僅如此，在將多重輸出(Multiple-Output)時域反應資料以單列或單行排列方式建立訊號矩陣的情形下，組合後的矩陣其各奇值平方將與組合前各矩陣相對應奇值的平方和（即能量總和）成正比，這表示各模組訊號可藉不同量測點的資料提升其強度。由於模組振幅係數可由振形推導而得，因此上述的結果可以提供使用者選擇較佳的輸出入點以提高訊雜比(Signal to Noise Ratio, SNR)，並能經此計算出更準確的模態參數。

關鍵字

奇值分解法 ； 模態參數 ； 訊雜比

並列摘要

Singular Value Decomposition has been widely used in the identification of dynamic system parameters. SVD is numerically stable and accurate but its computation is quite time-consuming. The uncorrelated components in a signal can be separated and stored in the singular vectors and singular values, where the former contains oscillating information and the latter represents power. This paper shows that the singular values are proportional to signal amplitudes and inversely proportional to natural frequency and damping ratio under appropriate conditions. If multiple outputs are placed orderly to construct the data matrix, it can be shown that the power of each mode is the sum of the power in each output, reflected in the singular values of the data matrix. As a result, the use of multiple outputs on the same data matrix is capable of promoting the average Signal to Noise Ratio (SNR) of modal parameters for a vibrating system. The modal amplitudes of targeted modes are therefore essential to the success of lowering SNR threshold. Since they may be found from mode shapes, the results of this research may be helpful in providing information for the selection of appropriate sensing and exciting degrees of freedom.

並列關鍵字

SVD ； modal parameters ； SNR