It is well known that Welch´s method includes bias and random errors and that these errors are contradictory. A trade-off between bias and random errors thus has to be made, given that a certain amount of data is available. Recently [5] investigated this thoroughly, and the main outcome of their paper, of interest in the present paper, is that the minimum errors (bias and random) using Welch´s method are obtained by using a half-sine window and 67% overlap processing instead of the more traditionally advocated Hanning window with 50% overlap. Alternative techniques for computing FRFs such as Danielle´s method, or “smoothed periodogram method”, and the original Blackman-Tukey method, where correlation functions are computed followed by applying FFT, are also well known and described in for example [3]. However, since using Welch´s method is an established “de facto standard” in commercial analysis systems, we will focus on this method for estimating FRFs in the comparisons below. In [4] it was shown that the random decrements (RD) technique can be used to compute frequency responses from impact testing. The RD technique offers an alternative to the above mentioned techniques for FRF estimation where the basic idea is in line with the Blackman-Tukey method, but where the estimation procedure is buffer oriented like Welch. In the following, a method using RD on systems excited by random forces is introduced and compared with the traditional technique using Welch´s method. WELCH´S METHOD We will assume that the signals exciting a linear system are random. Welch´s method is based on dividing the total data into a number of overlapping segments, applying a time window to each segment, and then computing the DFT of each windowed segment. When these steps have been taken, the spectral densities are estimated by ( ) ( ) 1 ( ) ˆ ( ) ( ) 1 ( ) ˆ ( ) ( ) 1 ( ) ˆ ( ) ( ) 1 ( ) ˆ 1 * 1 * 1 * 1 * k K k k YY k K k k YX k K k k XY k K k k XX Y Y K G Y X K G X Y K G X X K G (1) where ( ) kX is the DFT result of the k-th time windowed segment ( ) x t k . Frequency Response Function (FRF) estimates ( ) ˆ 1 H and ( ) ˆ 2 H are estimated from ( ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ 2 1 YX YY XX XY G G H G G H (2) 498 A good alternative to the above mentioned solutions to the bias problem is to use random signals and then reducing the bias by estimating RD functions and afterwards estimating the Frequency Response Functions (FRFs) from the RD functions. In As mentioned in the introduction, estimates in Eq. (2) are affected by bias as well as random errors. For analysis around a resonance of a mechanical system, the bias error is in principle dependent on the ratio of the resonance bandwidth and the frequency increment [3, 2], f Br '/ , where r r r f B 9 2| , and it is highest at the frequency line closest to the resonance frequency. The normalized random error is more complicated. In the case used in the present paper with no contaminating noise, however, as shown in [5], the random error is minimized by using the half-sine window with 67% overlap processing.
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