6 Towards Dynamic Substructuring Using Measured Impulse Response Functions 77 time domain frequency domain fh(t) uh(t) fh(ω) uh(ω) y(t) y(ω) FFT h IFFT finv ∗ u Fig. 6.2 Workflow of IRF determination; through the frequency domain or directly in the time domain 6.3.1 Frequency-Domain Approach In the frequency domain the first two tasks can be performed in a single step, typically by dividing the complex FFT spectra of the response and force signal. The relation for a single impact measurement his then found to be: yh.!/ D uh.!/ fh.!/ (6.9) In addition, the averaging step for Nh impacts is implemented by considering the averaged auto-power spectra (APS) and cross-power spectra (CPS) of the input and output signals, normally computed using the Fast Fourier Transform (FFT): y.!/ D Gfu.!/ Gff .!/ with 8 < : Gfu.!/ , 1 Nh Ph fh.!/ uh.!/ Gff .!/ , 1 Nh Ph fh.!/ fh.!/ (6.10) It can be shown that Eq. (6.10) performs least-squares averaging on the input spectra fh and provides the so-called H1 estimator for the FRF [1]. To obtain now the time-domain IRF, one simply computes the inverse Fourier transform of the averaged FRF using an IFFT. The whole procedure is depicted in Fig. 6.2. Although the frequency-domain approach as summarised is easily implemented, it has certain shortcomings: – The phase of the estimated FRF is fully determined by the phase of the averaged CPS Gfu.!/. This does however not guarantee that the resulting IRF is causal. Instead it is often experienced that the computed IRF shows a rapidly growing oscillation near the end of the time response. – FRF determination uses FFTs of the measurement blocks. If the response has not died out completely within the length of the measurement block, windowing should be applied to prevent spectral leakage. This is cumbersome as windowing adds artificial damping to the response. As we are seeking for a method to perform the abovementioned steps in order to obtain the IRF y.t/ rather than the FRF y.!/, it seems natural to seek for a solution in the time-domain as well. However this is not trivial; it can be understood from the division in Eq. (6.9) that the time-domain counterpart involves a deconvolution of u.t/ withf.t/, as was already reported in [9]. Direct deconvolution is known to be a delicate procedure, highly prone to measurement noise. It is there suggested to follow an alternative route, which is presented next. 6.3.2 Time-Domain Approach Let us recall Eq. (6.1a) and write the time-discretised convolution product for a one-dimensional case1 in a similar way as Eq. (6.4a). The indexing of the equations is adjusted such that y1 holds the response to the unit impulse at t1 D0. 1For now, displacements IRFs are considered in the derivation.
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