178 J.P. Noël et al. Table 16.4 Experimental clearances and stiffness parameters of the WEMS piecewise-linear springs given through a dimensional values for confidentiality reasons Neg. clearance Pos. clearance Linear stiffness Neg. nonlinear stiffness Pos. nonlinear stiffness X1 1.90 – 0.70 26.76 – X2 – 1.93 0.75 – 46.23 X3 – – 0.77 – – X4 – – 0.55 – – Y1 – – 0.82 – – Y2 – – 0.70 – – Y3 1.90 – 0.58 26.76 – Y4 – 1.93 1.02 – 46.23 Z1 1.01 1.55 8.30 118.07 79.40 Z2 0.84 1.62 9.21 116.73 88.41 Z3 0.93 1.59 9.18 118.07 79.40 Z4 0.93 1.59 10.03 116.73 88.41 5 10 20 30 40 50 0 10 20 30 40 50 60 Frequency (Hz) Model order Fig. 16.7 Stabilisation diagram computed by the FNSI method using three measured channels. Cross: stabilisation in natural frequency; square: extra stabilisation in damping ratio; circle: extra stabilisation in MACX; triangle: full stabilisation. Stabilisation thresholds in natural frequency, damping ratio and MACX value are 2 %, 5 % and 0.98, respectively 16.6.2 Activation of a Single Nonlinearity of the WEMS Device A multisine with a flat amplitude spectrum in 5–50 Hz was applied in the axial direction to NC 2 on the inertia wheel side. Time integration was carried out over 30 periods of 8,192 samples using a nonlinear Newmark scheme considering a sampling frequency of 20,000 Hz. Time series were subsequently decimated down to 1,000 Hz. The amplitude and the location of the excitation caused axial impacts exclusively in NC 2. The level of Gaussian noise added to the synthetic signals was set to 2 % of the RMS amplitude of the axial response at the inertia wheel node. Similarly to the analysis of the Duffing oscillator in Sect. 16.5, three periods were rejected to avoid transient distortions, and five periods were kept for validating the subspace and maximum likelihood models. The average of the 22 remanning periods eventually yielded an estimate of the covariance matrix of noise corrupting each channel. The first step toward formulating a nonlinear subspace model of the SmallSat dynamics is the selection of an adequate model order, which translates the number of linear modes excited in the output data [5]. In linear system identification, stabilisation diagrams are most frequently exploited as decision-making tools and have proved successful in numerous industrial applications. A distinct advantage of the FNSI method is that it still allows the use of the stabilisation diagram for retrieving linear system parameters from nonlinear data [6]. Figure 16.7 charts the stabilisation of the natural frequencies,
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