## 4 Nonlinear Identification Based on Random Excitation
### Identification of Offset Freeplay  
Another series of numerical experiments of offset freeplay were carried out with similar simulating parameters as in the previous numerical experiment. The excitation was generated a Gaussian white-noise signal of 60,000 points in 30 s and band-limited in 0-150 Hz repeating 5 times. The analysis was also conducted in terms of generalized coordinates. The identification problem is complicated by the existence of offset in freeplay; however, it is still in reach with utilization of the TNSI method considering some adjustments and skills.  
The selection of the mathematical forms \(g_{j}(t)\) is different from central freeplay because of the offset. The identification of the left limit and the right limit of the freeplay region are operated separately in two different sets of displacement ranges. Similar to the procedure of identifying central freeplay, the 1 st two sets of ranges is based on the boundary of relative displacement response, i.e., \([\min(q_{\mathrm{r}}),0]\) and \([0,\max(q_{\mathrm{r}})]\). The displacement ranges can likely be divided into \(l_{-}\) and \(l_{+}\) equally spaced intervals, respectively, where the subscripts \(-\) and \(+\) stand for _negative_ and _positive_. Given the \(j\)th negative interval \([d_{j-1,-},d_{j,-}]\), the nonlinear form is defined as  
\[g_{j,-}(t)=\begin{cases}q_{\mathrm{r}}-d_{j,-}&,\mathrm{if}\,q_{\mathrm{r}}<d_ {j,-}\\ 0&,\mathrm{if}\,q_{\mathrm{r}}\geq d_{j,-}\end{cases} \tag{17}\]  
Similarly, for the \(j\)th positive interval \([d_{j-1,+},d_{j,+}]\), the nonlinear form is  
\[g_{j,+}(t)=\begin{cases}q_{\mathrm{r}}-d_{j,+}&,\mathrm{if}\,q_{\mathrm{r}}>d_ {j,+}\\ 0&,\mathrm{if}\,q_{\mathrm{r}}\leq d_{j,+}\end{cases} \tag{18}\]  
and the nonlinear terms is the summary of \(g_{j,-}(t)\) and \(g_{j,+}(t)\).  
Figure 13 gives the magnitude of nonlinear force contributions identified by TNSI method when \(l=15\) and \(l=20\). Both of the results indicate that the left limit of the freeplay region is around \(-0.05^{\circ}\), whereas the right one is near to \(0.15^{\circ}\). The reconstruction of the nonlinear restoring force is plotted in Figure 14, indicating an accurate estimation of the inflection points in the offset freeplay. A same procedure is also repeated considering division of the narrowed ranges \([-0.07^{\circ},-0.03^{\circ}]\) and \([0.13^{\circ},0.17^{\circ}]\) to improve the results as the force contribution in Figure 15a and the nonlinear restoring force in Figure 15b. The modal parameters of OLS can still be extracted from the stabilization diagram in Figure 16.  
Figure 13: The identification of offset freeplay using the 1 st displacement range \([\min(q_{\mathrm{r}}),\max(q_{\mathrm{r}})]\).  
The identified natural frequencies of OLS based on the two numerical experiments previously are summarized in Table 3.  
Figure 16: Stabilization diagram of OLS of offset freeplay computed by TNSI method using the estimated FRF(\(H_{22}\)). (Dot: Not stable in frequency; Circle: Stable in frequency; Cross: Stable in frequency and damping).  
Figure 14: The nonlinear restoring force of offset freeplay under different divisions.  
Figure 15: The identification of offset freeplay using the 2 nd displacement range.