24 D. Di Maio et al. Generation of the NFRS Using the Experimental Modal Analysis This section presents an experimental modal analysis technique for evaluating the amplitude-dependent parameters of nonlinear FRFs. As introduced, many test engineers find the data analysis of nonlinear FRFs very challenging because of the few methods available to process FRF data with distortion caused by source of nonlinearity. Furthermore, assuming that modal parameters are evaluated, the synthesis of an FRF is not straightforward, and thus, the assessment of the modal parameters is left unchecked. The previous section addressed the problem of generating nonlinear FRFs, which can be used for synthesizing nonlinear FRFs from the EMA approach. EMA using the modified-Dobson method The experimental identification is carried out on FRFs that are measured under steady-state conditions and for reasonably well-separated modes to avoid mode coupling. The Dobson method [11] is a modal analysis technique that allows using one of the properties of the inverse of the FRF. It works by line-fitting the real and imaginary parts of the inverse of the FRF and using the line-fitting coefficients to extract the modal parameters. The method is not used as often for linear modal analysis because of the more powerful techniques developed during the past two decades. However, it proves the best identification method for nonlinear FRF analysis. The underlying principle is the following. Three Nyquist points identify a circle, the properties of which can be curve-fit to identify the dynamics of a linear system. The circle -fit method was one of the most used techniques in the past for this purpose. Unfortunately, the nonlinear vibrations transform the circle into something else, for the stiffness nonlinearity part of the circle disappears, as shown in Figure 4 see the grey hollow dots. The circle stops at a certain amplitude level and then resumes at lower amplitudes. The response amplitude jump causes this because of the stiffness nonlinearity used to generate a numerical nonlinear FRF. The smart analysis, which exploits the Dobson method, fixes two frequency points at the lowest amplitude (the linear one) and uses a third one that sweeps all the other frequency points. Hence, every triplet of frequency points generates an equivalent Nyquist circle or an equivalent linear dynamical system. The goal is to observe how the dynamics of a system evolves from linear to nonlinear vibrations. Hence, one can generate as many triplets as the frequency points measured. By applying the line-fitting to the triplets, one can evaluate the modal properties at that vibration amplitude, thus evaluating an equivalent linear system. By repeating the analysis process, one can observe how the natural frequency curve changes because of the amount of vibrations. The reader is encouraged to expand this quite brief description by reading the full paper [11]. Figure 5 shows the final step of the analysis, where the natural frequency curve is evaluated at every amplitude level of the nonlinear FRF. The black dots of the natural frequency curve indicate the analysis carried out using the frequency points from 45 to approx. 51 Hz, while the blue dots the other part of the FRF branch. Hence, equation (1) can be used for generating an NFRS. α(ω,X)= rA(X) ω2 r(X) −ω2 +iηr(X)ω2 r(X) Fig. 4 Nyquist circles for linear (blue hollow dots) and nonlinear FRFs (grey hollow dots) Fig. 5 Nonlinear FRF with the natural frequency curve
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