## 3 Shaking Table Test
### Experimental Results and Analysis
#### 3.2.2 Parameter Identification  
Many methods can be used to identify the equivalent damping ratio of the test model based on the single degree of freedom [21; 22; 23; 24]. To describe the damping ratio of the model more objectively, a parametric method based on multiple degrees of freedom [25] was used to calculate the constant damping ratio. In this method, the acceleration response is assumed to be composed of multiple vibration modes:  
\[Z_{p}(t)=\sum_{s=1}^{N}u_{\mathrm{ps}}\tilde{q}_{s}(t)+a(t) \tag{13}\]  
where \(u_{\mathrm{ps}}\) is the \(s\)th mode contribution; \(a(t)\) is the ground motion accelerations; and \(q_{\mathrm{s}}(t)\) is the relative displacement of the \(s\)th mode, which satisfies the following equation:  
\[\tilde{q}_{s}(t)+2h_{\mathrm{s}}\omega_{\mathrm{s}}\hat{q}_{s}(t)+\omega_{ \mathrm{s}}^{2}q_{s}(t)=-\beta_{\mathrm{s}}a(t) \tag{14}\]  
Assuming that the starting and ending points of the time histories are both zero, the Fourier transformation is performed on both sides of Equation (14):  
\[Z_{p}(\omega)=\left(\sum_{s=1}^{N}\frac{\omega^{2}\beta_{\mathrm{s}}u_{ \mathrm{ps}}}{\omega_{\mathrm{s}}^{2}-\omega^{2}+2h_{\mathrm{s}}\omega_{ \mathrm{s}}\omega\mathrm{i}}+1\right)\!A(\omega) \tag{15}\]  
where \(h_{\mathrm{s}}\) is the damping ratio, \(w_{\mathrm{s}}\) is the natural frequency, and \(\beta_{\mathrm{s}}u_{\mathrm{ps}}\) is the effective participation factor at position \(\mathrm{p}\). In the following sections, the effective participation factor is simply called the participation factor. The normalized error is defined as follows:  
\[J=\int_{\omega_{1}}^{\omega_{2}}\!\!\left|Z_{\mathrm{obs}}(\omega)-Z(\omega) \right|^{2}\!\mathrm{d}\omega/\int_{\omega_{1}}^{\omega_{2}}\!\!\left|Z_{ \mathrm{obs}}(\omega)\right|^{2}\!\mathrm{d}\omega \tag{16}\]  
where \(Z(\omega)\) is the Fourier transformation calculated by Equation (15), and \(Z_{\mathrm{obs}}(\omega)\) is the Fourier transformation of the observed responses. In the structural parameter identification procedure, the normalized error was used to determine the least-square fit of the modal responses to the observed responses in the frequency domain over a specific frequency band (\(\omega_{1}\), \(\omega_{2}\)). For base-isolated structures, the structural vibration is mainly in the first mode. Therefore, in most cases, \(N\) equals 1, and the acceleration of the shaking table and the top of the structures can be used for Fourier transformation. The constant damping ratio and participation factor were calculated. The frequency band (0.6, 1.4) was used for horizontal parameter identification, and the frequency band (3, 3.5) was used for vertical parameter identification. The identified parameters are shown in Table 3 and Figure 18.  
Figure 18: Horizontal parameter identification results: (**a**) test condition No. 22; (**b**) test condition No. 23; (**c**) test condition No. 24. The results show that the damping ratios of all with-asphalt tests are about 13%, while those of non-asphalt tests are less than 2%. This means that the springs cannot provide a sufficient damping ratio. At the same time, the errors and participation factors of multi-dimensional input conditions are significantly larger than those of horizontal, horizontal and vertical inputs. The reason for this is that when the rocking ground motions are input, the right side of the differential equation of motion has the effect of rocking ground motions. In this case, the differential equation of motion should be expressed as follows:  
\[\ddot{q}_{s}(t)+2h_{s}\omega_{\hat{q}_{s}}(t)+\omega_{s}^{2}q_{s}(t)=-\beta_{ s}[a(t)+\ddot{a}h_{i}] \tag{17}\]  
where \(\ddot{a}\) is the rocking acceleration of the shaking table, and \(h_{i}\) is the height of the \(i\)th floor. Equation (17) was used to replace Equation (14), and the above modal analysis procedure was repeated to obtain the modal parameters under multi-dimensional ground motions.