30 An Efficient Method for the Quantification of the Frequency Domain. . . 311 30.2.3 Approach 2: Analytical Uncertainty Propagation Based on Frequency Domain Estimator An estimator for the discrete Fourier transform of the linear combiner of windowed responses QFgw.!k/ DFgw.!k/ C" DH.!k/Ffw.!k/ (30.18) can be derived from the frequency response relationship for continuous infinite Fourier transforms directly in the frequency domain. The discrete Fourier transform of the linear combiner of windowed responses Fgw.!k/ is the measure of interest that could be obtained from real measured data. Also approach 1 and 3 are focusing on the virtual generation of this measure. QFgw.!k/ is an accurate estimator for Fgw.!k/, if the errors " are sufficiently small, which is the case for a small discrete time step ( t !dt), a large time frame length (t !1), and a well-chosen window function. Time discretization errors are assumed to be negligible. However, errors from an insufficiently large time frame length and an inappropriate window function influence strongly the results of approach 2. Hence, the following aspects should be taken into account for the application of the estimator according to Eq. (30.18): • The estimator is very inaccurate, if the compact finite support of the window functionŒts; te contains a notable time period related to the transient state of the structural system. The best estimations are obtained, if the time period defined by the support of the window function does not include the transient state, but the steady state. This is approximately equivalent to the interpretation of having a weakly stationary process, where it is assumed that the statistics of the excitation is time-invariant and applied since t D 1. • The shape of the window function is essential for the accurateness of the estimator. For the case of random vibration, a suitable window function goes smoothly to zero at the boundaries and is constant elsewhere. The tradeoff between sharpness at the boundaries and flatness elsewhere has been tackled by many researchers. Recommended window functions for random Gaussian responses and excitations are the Hann, Hamming, and tapered cosine window. However, for a very large time frame length, a rectangular window is recommended. • The energy of the windowed time signal has to be identical to the energy of the original signal. Appropriate scaling factors for different window functions can be found in [7, p. 144]. In the following, the estimator of Eq. (30.18) is reformulated to obtain a linear operator between time domain excitations and frequency domain responses similar to approach 1. For a multiple-input multiple-output system, the frequency response function is given by H.!k/ DAˆx D.!k/ ˆf T (30.19) at frequency !k for a linear time invariant structural system. D.!k/ 2 Cm m is a complex diagonal matrix. Its diagonal elements .D.!k//l;l D . /l !2 k # 2!kp. /l . /l . /2 l 2!2 k . /l C!4 k C4!2 k . /l . / 2 l (30.20) depend on the circular frequency!k. A reformulation of Eq. (30.18) leads to the estimator of the discrete Fourier transform QFgwi .!k/ D mf X jD1 .H.!k//i;j 01 k 1 01 mw 2 k Bqwfj (30.21) of the ith linear combiner for a certain circular frequency!k with QFgwi .!k/ 2 C1 and qw D 0mw s 1 wImw ; (30.22) where Imw represents the identity matrix of dimension mw. The vector wis assembled by the discrete values of the support values of the window function w.t/ as defined in Sect. 30.2.2. The matrix 0mw s 1 is a zero valued matrix of dimension mw s 1. The expression qwfj is the windowed excitation at the jth degree of freedom and 01 k 1 1 01 mw 2 C1 k B represents rowk of the matrix of discrete coefficients B 2 C. mw 2 C1/ mw related to the Fourier transform as introduced in Eq. (30.12).
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