# Model-Free Identification of Nonlinear Restoring Force with Modified Observation Equation
## 3 Numerical Application
### Building Structure Equipped with SMA Damper  
In this example, an SMA damper rather than MR damper is employed to investigate the performance of the proposed approach for the identification of NRF around the unsmooth points. The NRF provided by SMA damper in a double-flag form as shown in Figure 4 can be given by the following equation:  
\[\mathbf{R}_{SMA}^{non} = \left\{ \begin{array}{ll}k_{SMA}^{1}\times s&(\alpha ab,o^{\prime} a^{\prime}b^{\prime})\\ (k_{SMA}^{1}-k_{SMA}^{2})\times s_{b}\times\text{sgn}(s)+k_{SMA}^{2}\times s&( bc,b^{\prime}c^{\prime})\\ (k_{SMA}^{1}-k_{SMA}^{2})\times(s_{b}-s_{c})\times\text{sgn}(s)+k_{SMA}^{1} \times s&(cd,c^{\prime}d^{\prime})\\ (k_{SMA}^{1}-k_{SMA}^{2})\times s_{a}\times\text{sgn}(s)+k_{SMA}^{2}\times s&( da,d^{\prime}a^{\prime})\end{array} \right. \tag{28}\]  
where \(k_{SMA}^{1}\) and \(k_{SMA}^{2}\) are the stiffness provided by SMA in different stages as shown in Figure 4; sgn(\(\cdot\)) is the sign function; \(s\) is relative displacement of the damper as mentioned before. The definition of \(s_{a}\), \(s_{b}\), \(s_{c}\) and \(s_{d}\) can be found in Figure 4. It can be seen that the stiffness as well as NRF provided by SMA damper at these points has a sudden change. Here, the SMA damper was assumed to be located on the second floor of the aforementioned six-story building, and a random excitation was applied to the top floor of the structure. The following parameters were selected for SMA damper: \(k_{SMA}^{1}\)= \(1.0\times 10^{5}\) N/m, \(k_{SMA}^{2}\) = \(0.5\times k_{SMA}^{1}\), \(s_{a}\) = \(4\times 10^{-3}\) m and \(s_{b}\) = \(8\times 10^{-3}\) m. The acceleration responses were measured with the sampling frequency of 1000 Hz for the identification. Here, the quantities to be identified included the extended state vector \(\mathbf{Z}=\left[x_{1},\dots,x_{6},\dot{x}_{1},\dots,\dot{x}_{6},k_{1},\dots,k_{6 },\alpha,\beta\right]^{T}\) and the NRF provided by SMA damper. The initial values of the unknowns were set to be 50% of the corresponding actual ones. Figure 3: Comparison of the identified MR damper force: (**a**) Hysteretic loop; (**b**) Time history of nonlinear restoring force (NRF). Figure 2: The identified stiffness (building with MR damper): (**a**) first floor; (**b**) third floor. By using the proposed approach, the unknown structural parameters could be identified as shown in Table 2. Obviously, the maximum error of the identified parameters was very small. Similarly, only the stiffness of the second and fifth floor is given in Figure 5 as examples. It can be clear that the prompt and stable convergence process can be obtained. Besides the identification of structural parameters including stiffness and damping coefficients, the NRF provided by SMA damper can be simultaneously identified as well. Moreover, to demonstrate the effectiveness of the proposed approach around the unsmooth points, another nonparametric NRF identification method basing on power series polynomial model (PSPM) is employed [24; 25]. The identified results by means of the proposed approach and PSPM method are plotted in Figure 6. It can be found from Figure 6a that more accurate results can be obtained by using the proposed approach especially around the unsmooth points. Figure 6b gives the compar \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Parameters** & **Actual** & **Identified** & **Error (\%)** & **Parameters** & **Actual** & **Identified** & **Error (\%)** \\ \hline \(k_{1}\) (kN/m) & 180.00 & 179.92 & \(-\)0.05 & \(k_{5}\) (kN/m) & 180.00 & 179.94 & \(-\)0.04 \\ \(k_{2}\) (kN/m) & 180.00 & 179.88 & \(-\)0.06 & \(k_{6}\) (kN/m) & 180.00 & 179.83 & \(-\)0.10 \\ \(k_{3}\) (kN/m) & 180.00 & 179.84 & \(-\)0.09 & \(\alpha\) & 0.2644 & 0.2643 & \(-\)0.04 \\ \(k_{4}\) (kN/m) & 180.00 & 179.90 & \(-\)0.06 & \(\beta\) & \(2.578\times 10^{-3}\) & \(2.591\times 10^{-3}\) & 0.55 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The identified results of building structure equipped with SMA damper. Figure 4: NRF provided by shape memory alloy (SMA) damper.