## 2 Methodology
### Experimental Methodology
#### 2.1.1 Nonlinear System Identification  
The transient MiniWoT experimental responses considered in this work exhibit moderately nonlinear stiffness effects and highly nonlinear induced damping; due to these characteristics, the system identification problem is not trivial. Many nonlinear system identification techniques have classically been proposed and used, e.g., based on the Volterra series or block-structured nonlinear models consisting of interconnected linear and nonlinear subsystems [14]. Due to the nature of the problem at hand, piecewise linear modelling is adopted here. Such approaches assume time-localised linear behaviour, allowing the application of well-known linear techniques to the short sections of the time-domain response of the studied nonlinear system. The piecewise linear system can be found using the fitting techniques, which minimize the distance between the signal and chosen fitting functions in a least squares sense. Such methods belong to a class of Nonlinear Least Squares (NLS) fitting techniques and have been successfully used in the past for the analysis of one dominant frequency component in beam sloshing experiments [12]. We extend the NLS analysis here to cover all major frequency components of the transient signals, and a description of the method as applied in this work follows. Let \(F1\) be the natural frequency of vibration of the first bending mode of the MiniWoT structure, as identified by studying the small amplitude vibrations of the structure. The time-dependent acceleration data \(y(t)\) contains approximately 75 samples of data per one period of motion \(T1=1/F1\). The size of the data bin is chosen to be \(N_{W}\approx 150\) samples, corresponding to 2 periods of motion of the first vibration mode of the structure. A classical damped multi-harmonic motion representation  
\[y_{e}=\sum_{j=1}^{M}\left[A_{j}\exp\left(-\zeta_{j}\omega_{j}t\right)\cos \left(\omega_{j}\sqrt{1-\zeta_{j}^{2}}t+\phi_{j}\right)\right] \tag{1}\]  
is fitted to the signal inside each data bin in a least squares sense, where \(M\) is the number of frequency components considered to be dominant in the step release response. Each signal component \(j\) is characterised by its amplitude \(A_{j}\), damping ratio \(\zeta_{j}\), frequency \(\omega_{j}\) and phase \(\phi_{j}\). Figure 2 exemplifies this methodology, showing the signal binning and decomposition. The black curve represents the original signal \(y(t)\). Starting from \(t=0\), a bin of length \(N_{W}\) is selected, an estimate \(y_{e}(t_{1})\) is obtained for bin number 1 using \(M\) signal components and then the bin slides by \(N_{S}\) samples, after which the procedure is repeated. The signal at time \(t_{k}\) will thus be described by the function \(y_{e}^{k}(t)\). The length of the sliding distance \(N_{S}\) establishes the time resolution of the analysis and is chosen such that each frequency component of interest can be studied at a sufficient level of detail while also keeping the computation relatively inexpensive. The value of \(N_{S}\) is exaggerated in Figure 2 for clarity; for the results presented in this work, \(N_{S}=4\) is used. The result of this analysis will identify the nonlinear system using the vectors \(A_{j}\), \(\zeta_{j}\), \(\omega_{j}\) and \(\phi_{j}\) of size \(N\), where \(N\) is the number of fitted functions \(y_{e}\) over all time bins. Since this nonlinear identification procedure is developed for transient signals, different signal components \(j\) will be attenuated at different rates characterised by \(\zeta_{j}\). The main challenge of this type of analysis is to ensure its robustness while multiple frequency components gradually disappear from the signal \(y(t)\). The problem that arises is that the basic form of the curve fitting algorithm will try to fit _all frequency components at all times_, even though the respective components may not be meaningfully present in the signal anymore due to their drop below the noise floor.