24 L. Renson et al. The controllers are implemented on a BeagleBone Black fitted with a custom data acquisition board (hardware schematics and associated software are open source and freely available [22]). All measurements are made at 5 kHz with no filtering. Estimations of the Fourier coefficients of the response, base displacement, and control action are calculated in real time on the control board. However, this was for convenience rather than a necessity. 3.4 Experimental Results 3.4.1 Preliminary Tests The forced response of the nonlinear oscillator is studied using the CBC algorithm detailed in Sect. 3.2.1. Recorded Fourier coefficients were averaged over ten samples, and each modification of the target coefficients X? was followed by maximum 10 waiting periods of 0.4 s each to let the transients die out. Fourier coefficients were assumed stationary if their absolute and relative variance was lower than5 10 4 and1 10 7, respectively. Starting from rest, the target coefficient A? 1 was initially increased by 0.05 mm to overcome the shaker stiction. The following amplitude increments were 0.02 mm. The tolerance on Eq.(3.9) was set to 5 10 4 for each Fourier coefficient and a maximum of 25 iterations was allowed to reach convergence. Figure 3.2a (solid line) shows the response amplitude of the oscillator in function of the base displacement for an excitation frequency of 20.9 Hz. Despite the good convergence of the fixed-point iteration at all measured points, Fig. 3.2b clearly shows that the base excitation contains a higher-harmonic content that can represent more than 20 % of the fundamental component amplitude. The regions with the largest higher-harmonic content were found to coincide with the fold regions where the base displacement is minimum. The presence of these higher harmonics is therefore attributed to friction forces in the shaker, which was also found to introduce small-amplitude higher harmonics when it was tested independently. The harmonic forcing procedure described in Sect. 3.2.3 was used to retrieve the harmonic base excitation of interest. Equations (3.12)–(3.13) are solved such that the total amplitude of the higher harmonics is limited to 1.5 % of the fundamental component. The excitation was corrected for the second, third and fourth harmonics only. For the Jacobian matrix calculation, the Fourier coefficient perturbation was 1.5 10 2 mm, and only two or three Jacobian matrix calculations per forcing frequency were sufficient to ensure the proper resolution of Eqs. (3.12)–(3.13). The maximum number of iterations was limited to 3. The oscillator response is significantly modified by applying this procedure as illustrated by the dashed line in Fig. 3.2a. The major source of error remains now limited to the first few points with very small base displacements. For the rest of our experimental investigations, the harmonic forcing procedure presented in Sect. 3.2.3 is always applied between the steps 4 and 5 of the methodology of Sect. 3.2.1. For the backbone identification, the procedure is also applied after the convergence of the phase quadrature condition. However, its effect on the objective function value was found to be minor such that it never required any further modification of the forcing frequency. 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 2.5 Base displacement (mm) Response amplitude (mm) 0 0.5 1 1.5 2 2.5 0 20 40 60 80 100 Response amplitude (mm) Relative higher−harmonic amp. (%) a b Fig. 3.2 Amplitude sweep performed on the nonlinear oscillator at 20.9 Hz. (a) Oscillation amplitude in function of the base displacement amplitude. (b) Evolution of the higher-harmonic content normalized by the amplitude of the fundamental component. Solid (dashed) line shows the test performed without (with) the harmonic forcing procedure presented in Sect. 3.2.3
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