178 D. Roettgen et al. Table 24.1 Linear modal parameters Mode fn (Hz) ζ (% cr) Shape description 7 129.6 0.397% 1st soft bending of Beam 8 172.6 0.322% 1st stiff bending of Beam 9 385.8 0.069% (2,0) Ovaling model 10 391.9 0.083% (2,0) Ovaling model 11 551.6 0.278% Axial mode 12 945.4 0.413% (3,0) Ovaling model 13 1025.7 0.076% 2nd soft bending of Beam Fig. 24.5 Model process overview Fig. 24.6 Time domain modal force and acceleration 24.4 Nonlinear Modal Model Extraction Overview In this section, the results from a traditional windowed sinusoid excitation are analyzed resulting in a nonlinear pseudo-modal model. The procedure used to identify nonlinear pseudo-modal models from experimental data as described in Sect. 2.1 can be broken into three main steps. First, a high-level excitation must be applied, such as the windowed sinusoid discussed in Sect. 2.3. Next, a single degree-of-freedom modal response must be obtained, often with the use of a spatial modal filter. Finally, using this modal response the restoring force surface technique described in Sect. 2.1 can be used to identify a nonlinear pseudo-modal model that best fits the measured data in a least squares sense (Fig. 24.5). For a baseline, the CPB was tested using a wide-bandwidth windowed sinusoid. For this initial test, the structure was excited using a windowed sinusoid where the center frequency, fe, was the first linear natural frequency of 130 Hz, and the excitation bandwidth, Δfr,was ±30 Hz. This provides a wide pulse of energy in the frequency domain near the first bending resonance of the structure. The voltage level was tuned to maximize the voltage output of the amplifier without exceeding the electrical limits. After applying the modal filter, the accelerometer measurements were transformed into a single DOF response of the first bending mode. Figure 24.6 shows the applied modal force and response in the time domain, while Fig. 24.7 shows the data in the frequency domain. Following the RFS procedure from Sect. 2.1, a nonlinear modal model was extracted using a cubic and quadradic polynomial spring and damper as the nonlinear forcing elements. A simulation of the model was performed and obtained good agreement with the measured response data as shown in Fig. 24.8. Early in time, at high amplitude levels, the signals have great correlation, while later in time the measurement has more damping and the model begins to overpredict. This wide bandwidth windowed sinusoid is similar to those applied to the system in recent works [4], where results with similar
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