For example, applying Eq. (10) on the coherence estimate in Fig. 2 e) which has a dip of approximately 0.87, would give an efficient coherence value of 2ˆ (10) 0.999 a indicating that the bias is almost entirely eliminated. This coherence function is giving the amount of contaminating noise remaining to bias the FRF estimate. It can also be used to calculate how many averages are needed for sufficient suppression of the input noise. MIMO ESTIMATOR In the case of MIMO estimation, the principle of periodic excitation is more complicated because the input cross-spectral matrix needs to be invertible [7]. The so-called “periodic random” excitation signal is the most common solution to this problem. In effect, this method is based on producing independent random signals with N samples for each shaker, and repeating these signals until the system is achieving a periodic steady-state response. After this waiting time, single records of all input and output signals are acquired and auto and cross-spectra accumulated. New independent excitation signals are then produced, which are repeated until steady-state conditions occur, after which a record is acquired and spectra accumulated, etc. Theoretically, at least as many realizations of the input signals as there are shakers need to be used, but in practice many more averages have to be used to produce suppression of output noise. For the time domain averaging procedure suggested in the present paper, a slight modification to the frequency domain approach has to be made. The difference is that for each realization of the random input signals, a number of blocks are acquired from the steady-state response. These blocks are time averaged before spectra are computed and accumulated in the standard frequency domain averaging process to produce the input cross-spectral matrix xx G and the input-output crossspectral matrix yx G . Since in this case frequency domain averages exist, the coherence (in this case, of course, multiple coherence) can be computed using standard procedures [7], although due to the few averages used, the estimate will be poorly defined. SIMULATIONS To illustrate the effect of time domain averaging, a 2DOF system with natural frequencies of 10 and 15 Hz and relative damping of 2 % was used. The output displacement of DOF 1 was computed by the time domain forced response technique described in [8], which can also be found in [7], and contaminating noise added to the input and output signals. To obtain a realistic case with low SNR around the resonances, the input pseudorandom noise was created with PSD approximately equal to the inverse of the frequency response and the contaminating noise with flat PSD. The SNR of the input and output signals are shown in Fig. 2 a) and b). In the first simulation, pseudo random noise was generated and 100 averages used, after waiting 10 periods for steady state conditions. The driving point FRF in DOF 1 was computed using both “traditional” frequency domain averaging, and using the procedure described in the present paper. The results are shown in Fig. 2. As can be seen in the figure, the time domain averaging decreases the bias error around the resonance, whereas both methods work well to remove the bias at the antiresonance. Thus time domain averaging is a powerful tool to decrease bias in the FRF estimate in cases where there is poor SNR around the resonance frequencies, which sometimes happens on weak structures which are difficult to excite near the resonances. In the second simulation, both DOFs were excited by periodic random noise. First, frequency domain periodic random was simulated with 10 idle periods followed by an acquired record, and in all 40 such records were averaged, thus comprising a total of 40 times 11 blocks of data. After this, new simulation data were generated for time domain averaging. The difference with this method is that after the idle blocks, 100 averages were acquired for each generated random block, and five such sequences were accumulated to compute the input and output cross-spectral matrices. In total, thus 110 times 5 blocks were used in the simulation. The results of this simulation are shown in Fig. 3, where the FRFs zoomed in around the first resonance and around the antiresonance are shown. As can be seen in the figure, the time domain method decreases the bias error due to the input noise at the resonance, whereas both methods perform well to suppress the effect of output noise at the antiresonance. 303
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