3 Experimental Analysis of a Softening-Hardening Nonlinear Oscillator using Control-Based Continuation 25 Fig. 3.3 Forced response of the SDOF shown in Fig. 3.1. (Filled circle) Amplitude of steady-state periodic responses measured during a series of amplitude sweeps. (Blue surface) The complete forced-response surface obtained using Gaussian process regression. (Red solid line) Backbone curve measured using CBC 3.4.2 Forced Response and Backbone Curve Repeated amplitude sweeps for fixed values of the forcing frequency were carried out between 20.5 and 21.35 Hz in steps of 0.05 Hz. CBC parameters are identical to those used in Sect. 3.4.1. At each data point, full time series measurements were made. These are shown as black dots in Fig. 3.3 where the forcing frequency and forcing amplitude (in mm) are plotted against the response amplitude. To aid visualisation, a continuous surface constructed from the individual data points is plotted in blue. This surface was created using Gaussian Process regression on the collected data points where the hyperparameters for the Gaussian process are calculated by maximizing the marginal likelihood of the hyper-parameters [23]. Due to the softening-hardening character of the system, the surface presents four fold regions between 20.55 and 21.15 Hz. The CBC algorithm was also used to extract the backbone curve of the nonlinear oscillator. The objective function (3.11) was defined using the phase difference between the base displacement and the strain gauge measurement. The tolerance on (3.11) was 5 10 3 rad, and the minimum frequency step was 10 4 Hz. The backbone curve was then discretized using a constant amplitude step h D0:02mm. The frequency was adapted as described in Sect. 3.2.2. The backbone curve measured using CBC and the phase quadrature condition introduced in Sect. 3.2.2 is superimposed (in red) to the nonlinear oscillator forced response in Fig. 3.3. The total experimental time required to generate the curve was 71 min for a total of 68 points. The measured backbone curve is also presented in a forcing frequency—response amplitude plot in Fig. 3.4. The system presents a softening characteristic up to 1.3 mm where the oscillation frequency has dropped by 3 %. At 1.3 mm the backbone curve presents a turning point, above which the system presents a hardening characteristic with a resonance frequency increasing from 20.6 to 21.3 Hz in the [1.3–2.5] mm displacement range. At high-amplitude, the fundamental Fourier coefficients still contribute the most to the response. The second and third harmonics are the largest higher harmonics in the oscillator response. However, their relative importance compared to the fundamental component does not exceed 5 and 2 %, respectively, such that, in the present case, a single-harmonic excitation was sufficient to accurately reach quadrature and isolate the NNM motion. The variability of our experimental results is also investigated in Fig. 3.4. The results obtained for two different (consecutive) CBC runs are superimposed to the first one. Overall, the result repeatability is excellent. The CBC method offers means to verify and validate the quality of the experimental results. Besides the convergence of the objective function (3.11), the assumption of single-harmonic base excitation and the invasive character of the (PD) controller can be assessed. This verification is performed in Fig. 3.5 where the root-mean-square (RMS) values of three different time series are shown in function of the base displacement amplitude. RMS values were normalized by the RMS
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