## 4 Experimental Investigation
### Case 2: The Nonlinear System  
The results of this case study thus far indicate that the SCEM-UA algorithm is capable of parameter identification with incomplete measurements of the structural response and existing noise. In this section, the algorithm is applied to a nonlinear structural system, which is a realistic 5-story steel frame with a particle tuned mass damper (PTMD). The PTMD, combining the particle damping technology and TMDs, possess multiple damping mechanisms [34]. Considering the calculation efficiency, the simplified analytical method proposed by [PERSON] et al. [35] is selected to simulate the vibration reduction effects of the PTMD. The particles in the container are equivalent to a single particle and the schematic diagram of the primary structure with the PTMD as illustrated by Figure 15. The suspended container is considered as the (n+1)th DOF with an external force \(F_{\mathrm{P}}\) caused by the collisions between the container and the particle. Therefore, the governing equation of the primary structure can be rewritten as  
\[\left\{ \begin{array}{l}\mathbf{\dot{M}}\mathbf{\dot{x}}(t)+\mathbf{\dot{C}} \mathbf{\dot{x}}(t)+\mathbf{K}\mathbf{x}(t)=\mathbf{\ddot{E}}\mathbf{\ddot{x}}g+ \phi(c_{\mathrm{c}}\dot{y}_{1}+k_{\mathrm{c}}y_{1})\\ m_{\mathrm{c}}\ddot{x}_{\mathrm{c}}+c_{\mathrm{c}}\ddot{y}_{1}+k_{\mathrm{c}}y _{1}-c_{\mathrm{P}}H(y_{2},\dot{y}_{2})-k_{\mathrm{P}}G(y_{2})=0\\ m_{\mathrm{p}}\ddot{x}_{\mathrm{p}}+c_{\mathrm{p}}H(y_{2},\dot{y}_{2})+k_{ \mathrm{P}}G(y_{2})=0\end{array} \right., \tag{4}\]  
\[\mathbf{E}=\left[-m_{1}-m_{2}\ \dots\ m_{\mathrm{n}}\right]^{T}, \tag{5}\]  
\[\phi=\left[0\ 0\ \dots\ 0\ 1\right]^{T}, \tag{6}\]  
where \(\phi\) is the location vector of the control force, \(y_{1}=x_{\mathrm{c}}-x_{5}\), \(y_{2}=x_{\mathrm{p}}-x_{\mathrm{c}}\), and the container and the simplified particle are represented by the subscripts c and p, respectively. \(G(y_{2})\) and H(\(y_{2}\),\(\dot{y}_{2}\)) are nonlinear functions related to the clearance \(d\) of the equivalent single-particle damper, which can be obtained based on the equivalent principle. In addition, \(k_{\mathrm{c}}=\omega_{\mathrm{c}}^{2}m_{\mathrm{c}}\) and \(c_{\mathrm{c}}=2m_{\mathrm{c}}\xi_{\mathrm{c}}\omega_{\mathrm{c}}\) represent the interactions between the primary structure and the container. \(k_{\mathrm{p}}=\omega_{\mathrm{p}}^{2}m_{\mathrm{P}}\) and \(c_{\mathrm{P}}=2m_{\mathrm{P}}\xi_{\mathrm{P}}\omega_{\mathrm{P}}\) represent collisions between the container and the particles. More details about the equivalent principle and the simplified analytical method are given by [PERSON] et al. [35]. The PTMD with a 2.26% total auxiliary mass ratio to the primary system is suspended on top of the primary steel frame by four steel strands. The dynamic equation of the structure's system is discussed in Sections 3 and 4.1. The mass matrix of the steel frame \(\mathbf{M}=\) (1032 kg, 1032 kg, 1032 kg, 1032 kg, 829.9 kg), and the total masses of the container and the particles, \(m_{\mathrm{c}}=\) 39.345 kg and \(m_{\mathrm{P}}=\) 7.96 kg, are already known. In addition, based on the fundamental theorem of the structural dynamics, the stiffness and the damping coefficient can be determined by the circular frequency and the damping  
Figure 15: The test model and a simplified schematic of the system. ratio, respectively. Therefore, the parameters of the structure system, which are unable to be measured can be described by the parameter set  
\[\theta=\ \Big{(}k_{1},k_{2},\ldots,k_{5},\zeta_{1},\zeta_{2},\omega_{c},\zeta_{c}, \omega_{p},\zeta_{p}\Big{)}. \tag{7}\]  
In this study, the structure system response excited by the one-dimensional Shanghai artificial wave (SHW2, 1996) ground motion is selected as an example and the peak value of the acceleration is 0.05 g. Acceleration measurements at floors \(1,3,5\) are available in this case. The SCEM-UA algorithm was implemented with q = 20 complexes including 20 points in each complex.