88 H. Xiuchang et al. 0 a b 200 400 600 800 1000 10 -4 10 -3 10 -2 10 -1 10 0 10 1 frequency (Hz) amplitude (g/N) original SVD filtered 0 200 400 600 800 1000 1200 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 number of singular values normalized singular value Fig. 7.7 Original and filtered FRFs of raft (a) and the normalized singular values (b) 7.3.3 Truncated SVD to Eliminate Measurement Noise Although a lot of techniques such as averaging, filtering during the measurement and different FRF estimators for system identification (such as H1 and H2) have been used, it is impossible to perform noise-free measurements. If one wants to use the measured FRFs directly, the level of noise in the measured data determines the success of the analysis. Minimizing the noise content in FRFs after measurement is necessary to improve the quality of the measured data. The truncated SVD scheme is used to eliminate the measurement error. For a given FRFHij(!),which is composed of Ndiscrete frequency points, Hij(!) D[H1, H2 : : : HN] T, the corresponding Impulse Response Function (IRF) can be obtained through FFT. Suppose the IRF is hij(t) D[h1, h2 : : : hN] T, the Hackle matrix is formulated as Am n D 2 66 4 h1 h2 : : : hn h2 h3 : : : hnC1 : : : : : : : : : : : : hm hmC1 : : : hs 3 77 5 (7.2) where mCn 1Ds, Aij DhiCj 1. The SVD is performed ADUDV, where U and V are unitary matrices and D contains the singular values. By retaining the first r singular values with threshold d0, i.e., D(r,r)>d0D(1,1) and D(rC1, rC1)<d0D(1,1), the filtered Hackle matrix is constructed by the reduced unitary matrix A DUrDrVr. The ith element of the filtered IRF is hi D 1 k l C1 k Xj Dl Ai jC1;j .l Dmax.1; i mC1/ ; k Dmin.n; i// (7.3) where the length of his s, the filtered FRFH(!) is obtained by IFFT. The algorithm is carried out for one FRF of the raft. The thresholdd0 is chosen as 0.001. i.e., all the singular values which are larger than 0.1 % of the largest singular value are taken into account for each frequency line. The result is shown in Fig. 7.7. Generally speaking, it can be claimed that the noise has been smoothed out and FRF based on applying truncated SVD is significantly improved and of physical meaning. This is because the influence of measurement noise is reduced to some extent since the truncated SVD scheme can inherently deal with random errors.
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