OSE801 Engineering System Identification Lecture 09: Computing Impulse and Frequency Response Functions
1 Extracting Impulse and Frequency Response Functions In the preceding sections, signal processing using the Fourier transforms have been presented. In the remainder of this chapter, we will employ a set of sample problems to gain insight into both time and frequency domain methods to obtain the impulse and frequency response functions (IRFs and FRFs In so doing, both the fast Fourier transform(fft) and discrete wavelet transform (DWT) will be introduced, which will be studied in more detail in the subsequent lectures. The following materials are taken mostly from a thesis by Dr. A. Robertson who has graciously allowed the materials for class purposes. 1.1 Single Degree of Freedom Problem This first example is a simple one degree-of-freedom (DOF) spring-mass system that is modeled in the form: ẍ + ω 2 x = f(t), ω = 10π (1) and is subjected to a time-varying harmonic excitation, f(t). The input (forcing function ) and response of this system are shown in Figure 1. Two discontinuities, or spikes, were added to the harmonic input to accentuate the differences between the FFT and DWT algorithms. Theoretically, the Markov parameters (impulse response) of this example problem should be h(t) = A sin ωt (2) regardless of the input variations. Figure 2 shows the Markov parameters (impulse response functions) of the system as determined by both wavelet and FFT-based algorithms. Two FFTbased extractions have been presented: one with windowing and the other without. The DWT method picks up the Markov parameters almost exactly with only a small error at the end. The FFT-based Markov parameters without any filtering are off in magnitude and though they seem to determine the dominant frequency, leakage problems are imposing added frequencies to the representation (see Figure 3). A Hanning window was applied to this case to demonstrate the improvements and weaknesses of windowing procedures in the FFT method. The Hanning window improves the magnitude and representation of the Markov parameters, but also induces artificial damping to the system as seen in Figure 3. It is observed in Figure 3 that the frequency response functions (FRFs) constructed by the two FFT-based algorithms are also inferior to that of the 2
Fig. 1. Input and Response of One DOF System Fig. 2. Markov Parameters of One DOF System 3
Fig. 3. Frequency Response Functions of One DOF System wavelet algorithm for this system. While the frequency magnitude for the windowed FFT-based algorithm is accurate, induced artificial damping has greatly affected the shape of the curve. The poor results via FFTs are expected and can be attributed to two problems. First is the lack of a rich frequency spectrum for the input to the system. In spectral methods, one must divide by the FFT of the input, which in this case will be mainly zeroes due to lack of frequencies in the signal. This will result in numerical ill-conditioning of the problem making it difficult to find an accurate solution. The second problem is leakage, in this case caused by the abrupt change in frequency mid-way through the input and the two spikes. These problems are typical in spectral methods which generally use broad-band frequency excitation such as random noise instead. Wavelet methods, however, are not prone to these problems and can handle a variety of input functions as seen by the accuracy of the determined Markov parameters in this problem. Note that only the magnitudes of the FRF curves are shown. The phase diagrams, though not shown here, have discrepancies similar to those of the magnitudes. 4
Fig. 4. Four DOF Spring-Mass System 1.2 Four DOF Spring-Mass System A four DOF spring-mass system will now be examined to show a slightly more complicated application of the impulse response extraction. The system as shown in Fig. 4 has four inputs and four outputs, one at each of the masses. The masses and stiffnesses are 1 and 10000, respectively. 1.2.1 Harmonic Excitation For the first example, this system was excited by a combination of harmonic forces at each of the nodes as shown in Figure 5 One set of the corresponding Markov parameters and FRF curves for this problem is shown in Figs. 6 and 7. The Markov parameters obtained by the FFT-based algorithm have more error in magnitude than those obtained by the wavelet algorithm, i.e., 10 4 for the exact and wavelet methods versus 10 5 for the FFT. An examination of the FRF curves for the system corresponding to the fourth DOF input and fourth output reveals that the wavelet method is picking up the correct frequencies with the correct magnitudes. The FFT method can find the correct frequencies, but with a large error in the shape of the frequency curves, affecting the damping of the system. Similar results are obtained for the remaining FRF curves that are not shown here. Once again, harmonic inputs were used to excite the system to show the inability of spectral methods to handle limited frequency band excitation adequately. More general inputs will therefore be utilized in the next section to analyze the same system. 5
Fig. 5. Input and Response of Harmonically Excited Four DOF System 1.2.2 Random Excitation Next consider the same four DOF system, except now excited by a random input at all four nodes (see Fig. 8). Figure 9 shows that the Markov parameters determined via the WT method are slightly better than the FFT-based ones, though not significantly. As compared to the Markov parameters obtained from the sinusoidal input, the superiority of the random input for spectral methods is not apparent. When using random inputs, ensemble-averaging is a necessity to alleviate the noise found in the resultant Markov parameters. Figure 10 shows how when just five ensembles are averaged together, the results improve significantly. More ensemble-averaging of the random data will improve results further as will be shown more thoroughly in the next section. On the other hand, when harmonic excitations are used to determine Markov parameters (as in the previous section), ensemble-averaging does not improve accuracy. The foregoing three examples are indicative of typical challenges facing structural system identification, including input spectra, filtering of noises, sampling techniques, computational algorithms, and the accuracy of the impulse response or frequency response functions. These and additional aspects will now be studied in this course. 6
Fig. 6. Markov Parameters of a Harmonically Excited Four DOF System 7
Fig. 7. FRFs of a Harmonically Excited Four DOF System 8
Fig. 8. Input and Response of Randomly Excited Four DOF System 9
Fig. 9. Markov Parameters of a Four DOF System with 1 ensemble 10
Fig. 10. Markov Parameters of a Four DOF System with 5 ensemble 11