Activation of a single nonlinearity of the WEMS device

A multisine excitation with a flat amplitude spectrum and a root-mean-squared (RMS) amplitude of 100 N was applied in the axial direction to NC 2 on the inertia wheel side.

The excited band was limited to 5 – 50 Hz to encompass the linear modes of interest.

Time integration was carried out using a nonlinear Newmark algorithm with a sampling rate of 20000 Hz, and the time histories were low-passed filtered prior to being decimated down to 1000 Hz. The SmallSat response was simulated over 6 periods of 8192 samples, 5 periods of which were discarded to achieve steady-state conditions. The choice of a relatively low number of periods is explained by the unnecessary need to average the measurements in the absence of noise. The amplitude and the location of the excitation caused impacts exclusively on the negative and positive mechanical stops of NC 2. In the application of the FNSI method, the number of block rows i is set to 60, and measured frequency lines are processed in 5 – 250 Hz to capture the contributions appearing outside the input band due to nonlinearities. This completes the second step in Fig. 3.1.

The next step in the FNSI methodology is the determination of an adequate model order, i.e. step 3 in Fig. 3.1. In linear system identification, stabilisation diagrams are most frequently utilised as convenient decision-making tools [9, 88]. They have proved success-ful in numerous industrial applications, as for instance in Ref. [87]. In the presence of nonlinearities, the model order translates the number of excited modes of the underlying linear structure (see Section 2.5). Because the FNSI method succeeds in decoupling the estimation of the linear and nonlinear parameters, the use of stabilisation diagrams is still effective in nonlinear system identification for selecting the model order.

Processing of all measured channels

Fig. 3.4 charts the stabilisation of the natural frequencies, damping ratios and mode shapes of the structure computed at 100 N RMS using FNSI for model orders up to 60.

This analysis was conducted using all measured channels, i.e. 39 DOFs or 13 nodes, as output data. In the diagram, the MAC for complex-valued mode shapes (MACX), as defined in Ref. [113], is considered as the damping mechanisms in the SmallSat were shown to be highly nonproportional.

5 10 20 30 40 50

0 10 20 30 40 50 60

Frequency (Hz)

Model order

Figure 3.4: Stabilisation diagram computed using all measured channels. Dot: new pole;

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 are 1 %, 5 % and 0.98, respectively. Blue lines indicate the selected orders.

Fig. 3.4 shows full stabilisation of 11 modes at order 38. Note that in the case of mode 2 around 9 Hz and mode 6 around 31 Hz, full stabilisation symbols only appear at order 40. However, since this is tested between successive model orders taking as reference the lowest order, equal stabilisation is also achieved at order 38, which is therefore selected to avoid spurious poles. Table 3.4 lists the relative errors on the estimated natural fre-quencies and damping ratios together with the diagonal MACX values. The results in this table demonstrate the ability of the FNSI method to recover the modal properties of the underlying linear spacecraft from nonlinear data. One notes the very low MACX

value for mode 2 in Table 3.4, which is due to a poor distinction in the model between the deformed shapes of modes 1 and 2. The overall quality of the linear parameter estimates is confirmed in Fig. 3.5, where the driving-point frequency response function (FRF) of the system calculated at 1 N RMS is compared with the corresponding FRF synthesised using the FNSI method at 100 N RMS. The H1 FRF calculated at high level is also visible in this plot and highlights the importance of the nonlinear distortions affecting the SmallSat dynamics.

5 10 20 30 40 50

−115

−105

−95

−85

Frequency (Hz)

Amplitude (dB)

Figure 3.5: Comparison between the driving-point FRF at low level, i.e. 1 N RMS, where no nonlinearity is activated (in black) and the corresponding curve synthesised using the FNSI method at high level (in blue), i.e. 100 N RMS. The FRF calculated using the H1 estimator at high level is also superposed (in orange).

The complex-valued coefficients of the NC 2 nonlinearities are depicted in Fig. 3.6 (a – b). Their real parts show marginal dependence upon frequency, and their imaginary parts are three orders of magnitude smaller (see Table 3.5). These observations are signs of an accurate estimation. Table 3.5 gives the averaged values of the nonlinear coefficients, the relative errors and the ratios between real and imaginary parts in logarithmic scaling.

This table highlights the excellent agreement between the identified and exact coefficients.

In this identification case involving a single nonlinear connection, the time needed to solve the parameter estimation problem amounts to 159 s.

Mode Error on ω0 (%) Error on ǫ (%) MACX

1 0.11 -0.85 1.00

2 -0.08 0.92 0.04

3 0.04 -0.43 1.00

4 0.03 0.27 1.00

5 0.01 0.09 1.00

6 0.06 -0.68 1.00

7 -0.01 0.17 0.99

8 0.00 0.51 1.00

9 0.05 2.20 0.99

10 0.02 0.51 1.00

11 -0.01 0.38 1.00

Table 3.4: Relative errors on the estimated natural frequencies and damping ratios (in

%) and diagonal MACX values computed at order 38 using all measured channels.

5 10 20 30 40 50

100 115 130

Frequency (Hz)

Real part

(a)

5 10 20 30 40 50

80 90 100

Frequency (Hz)

Real part

(b)

Figure 3.6: Complex-valued estimates of the NC 2 – Z nonlinear coefficients in (a) negative and (b) positive displacement computed using all measured channels. Blue lines indicate the exact values of the coefficients.

NC Exact value Real part Error (%) Log₁₀ (real/imag.)

2 – Z (neg.) 116.73 117.38 0.56 3.33

2 – Z (pos.) 88.41 88.71 0.34 3.00

Table 3.5: Estimates of the NC 2 – Z nonlinear coefficients computed using all measured channels. Real parts averaged over 5 – 50 Hz, relative errors (in %) and ratios between the real and imaginary parts (in logarithmic scaling).

Processing of three measured channels

A second analysis of the same data set can now be carried out considering only three output sensors, namely the axial DOFs on both nodes of NC 2 and of the inertia wheel node. This corresponds to the more practical situation where the number of available channels is limited, and where the system responses are only recorded close to the nonlin-earity. The resulting stabilisation diagram constructed by the FNSI method is depicted in Fig. 3.7 and reveals fewer modes than in Fig. 3.4. Specifically, only modes 1, 4, 6 and 8 are now identifiable as they involve an axial motion of the WEMS device.

5 10 20 30 40 50

0 10 20 30 40 50 60

Frequency (Hz)

Model order

Figure 3.7: Stabilisation diagram computed using three measured channels. Stabilisa-tion thresholds in natural frequency, damping ratio and MACX are 1 %, 5 % and 0.98, respectively.

Full stabilisation is achieved at order 8, and the corresponding modal parameters are given in Table 3.6. One observes very accurate results, except in the case of the damping ratio of mode 8, which nevertheless consists predominantly in a rotation of the instrument panel. The updated FNSI estimates of the nonlinear coefficients are shown in Fig. 3.8 (a – b) and summarised in Table 3.7. The somewhat limited number of analysed sensors is found to yield a significant decrease of the computational burden (10 s), but deteriorates the accuracy of the nonlinear coefficients through the appearance of a drift over the frequency axis of their real and imaginary parts. More important peaks are also visible in their frequency dependence, translating in lower real-to-imaginary ratios in Table 3.7.

Mode Error on ω0 (%) Error on ǫ (%) MACX

1 0.10 -0.55 1.00

4 0.10 0.51 1.00

6 0.27 2.99 1.00

8 -0.37 -12.52 1.00

Table 3.6: Relative errors on the estimated natural frequencies and damping ratios (in

%) and diagonal MACX values computed at order 8 using three measured channels.

5 10 20 30 40 50

100 115 130

Frequency (Hz)

Real part

(a)

5 10 20 30 40 50

80 90 100

Frequency (Hz)

Real part

(b)

Figure 3.8: Complex-valued estimates of the NC 2 – Z nonlinear coefficients in (a) negative and (a) positive displacement computed using three measured channels.

NC Exact value Real part Error (%) Log₁₀ (real/imag.)

2 – Z (neg.) 166.73 116.47 -0.22 2.63

2 – Z (pos.) 88.41 88.96 0.63 2.60

Table 3.7: Estimates of the NC 2 – Z nonlinear coefficients computed using three measured channels. Real parts averaged over 5 – 50 Hz, relative errors (in %) and ratios between the real and imaginary parts (in logarithmic scaling).

Fig. 3.8 (a – b) demonstrates that the information contained in the frequency dependence of the nonlinear coefficients is particularly valuable to assess the quality of the identifica-tion. Specifically, inaccuracies in the estimation of the linear system parameters generally translate into peaks in the frequency-dependent coefficients. Missing modes and errors in the modelling of the nonlinearities rather affect the entire frequency dependence of the coefficients through the appearance of a drift.