## 3 SSI System versus Fixed-Base Oscillator  
### Dynamic Properties of SSI  
The simplest layered half-space, a single soil layer on bedrock, is taken for the analysis of the dynamic properties of the SSI system. The following parameters of the site are used: \(\rho_{\rm R}/\rho_{j}=1\), \(\
u_{\rm R}=\
u_{j}=1/3\), \(\xi_{\rm R}=\xi_{j}=0.02\), \(\beta_{\rm R}/\beta_{j}\)= 2, \(D/c=2\), (\(j=1\), 2, 3, \(\ldots\)). The foundation embedment is \(c/a=0.5\), and the foundation mass is \(M_{0}/M_{\rm s}=0.2\). Meanwhile, for the structure, we have a damping ratio \(\xi_{\rm b}=0.02\), a height \(H/a=2\), and mass \(M_{\rm b}/M_{0}=10\) following the recommended relationship \(M_{\rm b}/M_{0}=4.9H/a\) in [31]. The length-to-width ratio of the foundation varies, as \(b/a=1/4\), \(1/2\), \(1\), \(2\) and \(4\).  
Figure 3a shows the spectrum amplitude of structure response \(|u_{\rm b}|\) for a fixed-base oscillator and SSI systems with a varying \(b/a\) ratio. The \(x\)-axis is the dimensionless incident frequency \(\eta=0\)-0.3, which covers the predominant frequency domain of energy in earthquakes. Three structure frequency values are concerned: \(\eta_{\rm b}=0.1\), \(0.15\), \(0.2\). For a fixed-base oscillator, a rigid connection between the structure and the ground is assumed without any energy dissipation through the soil, while for an SSI system, the soil behaves like a cushion under the structure and adds additional flexibility to the system, so the system frequency of the SSI is obviously lower than the resonant frequency of the corresponding fixed-base oscillator. Further, as the length-to-width ratio \(b/a\) decreases, the peak has a higher frequency and a higher amplitude, approaching the fixed-base case.  
Figure 3b shows the dynamic properties (system frequency, system damping, peak amplitude) of the structure response \(|u_{\rm b}|\), as well as the varying \(b/a\) ratios. The \(x\)-axis is the dimensionless structure frequency \(\eta_{\rm b}\) = 0-3. The system frequency \(\widetilde{\eta}_{\rm b}\) corresponds to the peak frequency of \(|u_{\rm b}|\), or the eigen-value of Equation (1) [28; 29; 30], and here we use the ratio \(\widetilde{\eta}_{\rm b}/\eta_{\rm b}\) to indicate a shift in system frequency with respect to the structure frequency \(\eta_{\rm b}\). Although the spectrum shape is similar between the fixed-base oscillator and the SSI system with various length-to-width ratios, for the values of \(b/a\) and \(\eta_{\rm b}\) concerned in the figure, the system frequency shifts up to 50% from the structure frequency. For example, it shifts by 12% for a framed structure of width \(2a\) = 40 m, length \(2b\) = 40 m and fixed-base frequency \(\omega_{\rm b}\,=\,6.28\,\)Hz on a moderate rigid site (\(\beta_{1}\) = 300 m/s), and it shifts by 16% for the same structure on a moderate soft site (\(\beta_{1}\) = 250 m/s). In fact, the shift in peak frequency between the SSI system and the fixed-base oscillator leads to a completely different correspondence with respect to the frequency of excitations, which is further investigated in Figure 4.  
The second sub-figure of Figure 3b describes the system damping \(\widetilde{\xi}_{\rm b}\) which is obtained by the half-power method from \(\left|u_{\rm b}\right|\), and we use the ratio \(\widetilde{\xi}_{\rm b}/\widetilde{\xi}_{\rm b}\) to show a comparison between the SSI system and the fixed-base oscillator. The ratio \(\widetilde{\xi}_{\rm b}/\widetilde{\xi}_{\rm b}\ >\ 1\) indicates that the system damping is obviously larger than the structure damping, while the third sub-figure is the peak amplitude \(\left|u_{\rm b}^{\rm max}\right|\) between the SSI system and the fixed-base oscillator. Correspondingly, the peak amplitude in the frequency domain of the SSI system is smaller than the fixed-base oscillator, which gives \(\left|u_{\rm b\_SSI}^{\rm max}/u_{\rm b\_fixed\_base}^{\rm max}\right|\ <\ 1\).