## 2 Methodology
### Experimental Methodology
#### 2.1.2 Example of Miniwot Data Curve Fitting  
Before proceeding further, consider an example system identification result for one of the experimental cases studied in this work, showing the quality of the fitted results. Figure 3 shows an example of the acceleration signal at the tip of the fuel tank for a  
Figure 2: Binning of the time data and NLS fitting procedure.  
representative dry MiniWoT test. The original experimental data are shown in black; the reconstructed signal is shown in green, obtained by piecing together all of the \(y_{e}^{k}(t)\) signals in each time bin, as exemplified in Figure 2. A very good match is seen by the overlap of the two curves at all times. The insets also show details of the signal trace in time, indicating the damping of the higher frequency components at different rates until only the first bending mode is dominant.  
The suitability of the identification technique can also be quantified using the residuals. The residuals for each fitted function are computed as  
\[Res(y_{e})=y(t)-y_{e}(t) \tag{2}\]  
and the sum of squares of the residuals (SSR) in a bin is  
\[Res2(y_{e})=\sum_{i=1}^{N_{\text{W}}}\left(y(t)-y_{e}(t)\right)^{2} \tag{3}\]  
with the units of acceleration squared.  
An example of residuals for this analysis is shown in Figure 4 for the fitting case shown in Figure 3. Figure 4a shows the residuals for the first fitted function immediately after \(t=0\) for each sample inside the bin of length \(N_{\text{W}}\), while Figure 4b shows the SSR (summed squared residuals) values for all data bins, e.g., the first point represents the SSR for the bin shown in Figure 4a. Residuals of approximately 10% of the maximum acceleration value can be seen in the first fitted bin, and the squared residuals are dropping together with the amplitude of the signal, as expected. The fitting error is largest in the first cycle of motion due to the very rich frequency content following immediately after the step release event. Such values for the residuals are common for this type of piecewise linear analysis [15]. Correlating these results with the qualitative view in Figure 3, we conclude that this method is able to parameterise the damped transient responses accurately. Its utility for interpreting the studied processes will be illustrated in the following sections.  
Figure 3: Example of NLS identification result. Original signal is shown in black and the reconstructed signal using the fitted functions is shown in green.  
Before moving forward, it is important to discuss one possible limitation of the damping ratio identification _specifically_ relevant in extremely nonlinear circumstances. The damping ratio of a particular physical system may exhibit amplitude-dependent nonlinearities such that the values \(\zeta_{j}\) change substantially (nonlinearly) within the duration of a single data bin, i.e., one or two periods of motion in this case. Because of the linear response assumption within a single data bin (implying a constant \(\zeta_{j}\)), the choice of the bin size \(N_{W}\) can lead to the smearing and averaging effect on the \(\zeta_{j}\) values when traversing this region. Thus, the identified damping estimate may be weighted by the consecutive or neighbouring damping values in such highly nonlinear instances. Further, it was found that a data bin of the length less than the two periods of motion of the lowest frequency leads to substantial oscillations in the solution for the cases presented here. Consequently, we assume constant \(\zeta_{j}\) values over the two fundamental periods of motion. Here, as well as in other instances of time-frequency analysis, there is a compromise to be made between the time resolution of the analysis and the robustness of the damping estimation.