## 4 Nonlinear Identification Based on Random Excitation
### Identification of Central Freeplay  
As the TNSI method exploited directly _Input-Output_ data in time-domain that will not suffer from leakage and aliasing for no need to transform to frequency-domain, a random noise forcing was applied perpendicularly on the all movable fin at the same exciting location in Figure 4. The excitation signal consisted of 60,000 points over 30 s and band-limited in 0-150 Hz of interests, and repeated 5 times at each excitation level. It also has been discussed in [29] that the transient response was generally found to yield improved results compared with the steady-state regime, especially in the estimation of the linear modal properties. The analysis was conducted using the output data in terms of generalized coordinates established by the modal synthesis method for straightforward identification of freeplay. Therefore, the excitation force has to transform into generalized forces as input data. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type** & **Parameters** & **Value** \\ \hline Central freeplay & \(d\) & 0.10\({}^{\circ}\) \\ \hline Offset freeplay & \(d_{1}\) & 0.15\({}^{\circ}\) \\  & \(d_{2}\) & 0.05\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameters of freeplay. Figure 6: The finite element model of the all movable fin and elastic beam. However, the accuracy of nonlinear system identification is conditional upon the selection of the mathematical forms \(g_{j}(t)\) introduced in Equation (1). In particular, the boundary of different subsystem regions in Figure 2 plays a key rule in freeplay nonlinearity and still remain a challenging task. First, user-defined multiple piecewise-linear functions with varying clearances are used to representing the nonlinearity to be identified. Then, the freeplay can be posterior, determined after the reconstruction of nonlinear restoring force established by the participation of the user-defined functions. In a detailed explanation, we can assume the relative displacement across the nonlinear connection is \(q_{t}\). Then, the displacement range of \(q_{t}\), i.e., \([0,\max(|q_{t}|)]\), can be divided into \(l\) equally spaced intervals. Given the \(j\)th interval \([d_{j-1},d_{j}]\), the nonlinear form is defined as  
\[g_{j}(t)= \begin{cases}q_{t}-d_{j}\text{sign}(q_{t})&\text{, if }|q_{t}|>d_{j}\\ 0&\text{, if }|q_{t}|\leq d_{j}\end{cases}  \tag{16}\]  
where \(\text{sign}(\bullet)\) is the sign function. Figure 7 shows the nonlinear coefficients estimated by Equation (12). All estimations of \(d_{j}\) indicate that the limit of central freeplay lies at approximately \(0.10^{\circ}\). Then, the reconstruction of the nonlinear restoring force is performed based on the identified nonlinear coefficients \(\mu_{j}\) and mathematical form \(g_{j}(t)\), which is shown in Figure 8. However, it seems that dividing more intervals would not improve or even reduce the accuracy of identification, even though the estimated limits of the freeplay region are within an acceptable range. Therefore, a proper division of the displacement range is essential and the number of the interval \(l\) around 10 would lead a better estimation. Figure 7: The estimation of the nonlinear contribution of central freeplay using the 1 st displacement range \([0,\max(q_{t})]\). An alternative way to improve the accuracy is to narrow the displacement range of varying clearances and repeat a 2 nd estimation. A further division of the \(0.08^{\circ}\sim 0.12^{\circ}\) range into 10 new sub-intervals based on the 1 st estimation of nonlinear coefficient in Figure 7. The identified nonlinear coefficients in the range from \(0.08^{\circ}\) to \(0.12^{\circ}\) is presented in Figure 9a that a more precise estimate can be inferred, and the identified nonlinear restoring force is shown in Figure 9b, which meets the theoretical nonlinear restoring force with high consistency.