## 3 Isolation Effect  
### Solutions of the System  
The isolation effect of the proposed MDOF isolation platform with SLSs reflected by displacement transmissibility is based on the dynamic equation (Equation (12)) and the responses for different structural parameters can be obtained by solving the linearized dynamic equations. When vibration only exits in one direction, the natural frequency in this direction follows \(\sqrt{\frac{linear\ stiffness\ coeff\ cint}{equivalent\ mass}}\), e.g., \(\omega_{x}=\sqrt{\frac{2k_{i}\cos^{2}\phi\ \tan^{2}\theta}{Mn^{2}}}\) for \(z_{y}=z_{\varphi}=\hat{y}=\hat{\varphi}=0\); \(\omega_{y}=\sqrt{\frac{2k_{i}\sin^{2}\phi\ \tan^{2}\theta}{Mn^{2}}}\) for \(z_{x}=z_{\varphi}=\hat{x}=\hat{\varphi}=0\); \(\omega_{\varphi}=\sqrt{\frac{2k_{i}L^{2}\sin^{2}\phi\ \tan^{2}}{ln^{2}\theta}}\) for \(z_{x}=z_{y}=\hat{x}=\hat{y}=0\). While there are vibrations in all the three directions, the stiffness matrix in Equation (13) reveals that the three-direction vibrations are coupled in linear parts; thus, the anti-frequency band, where the amplitude of response is very small because of the anti-frequency point, occurs and the bandwidth of the bandgap is dependent on the adjustable structural parameters \(n\), \(\theta\), and \(\varphi\). When the excitation is harmonic excitation, such as \(x_{ei}=e_{i}cos\omega t\), the responses could be set as \(x_{i}=a_{i}\)cos(\(\omega t+\phi\)). Then, the displacement transmissibility \(T_{i}\) (dB) could be defined as the ratio between the amplitude of the response and the excitation in respective directions as:  
\[T_{i}=10\log_{10}\left(\frac{|a_{i}|}{|e_{i}|}\right)\left(i=x,y,z,\phi_{x}, \phi_{y},\phi_{z}\right) \tag{13}\]  
where \(a_{i}\) is the amplitude of response and \(e_{i}\) is the amplitude of the harmonic excitation in different directions. The dynamic equation (Equation (12)) and mass, stiffness, and damping matrices, reveal that the values of the components of the dynamic equations of each degree of freedom can be adjusted by changing the values of the structural parameters \(n\), \(\theta\), and \(l\). The isolation effects for different structural parameters are shown in Figure 8.  
Figure 8: Isolation effect in the \(x\) direction for different structural parameters: (**a**) different \(n\); (**b**) different \(\theta\); and (**c**) different \(l\).  
From Figure 8, it can be seen that, firstly, the natural frequency where has largest peak is reduced by increasing \(n\), or decreasing \(\theta\) and \(l\), which verifies the analysis of natural frequencies in the previous section. Secondly, the value of the peak is reduced by increasing \(n\) and \(l\), while changing \(\theta\) has no obvious effect on the value of the peak and, thirdly, the response of the amplitudes are lower for larger \(n\) and \(l\), or smaller \(\theta\). Additionally, it demonstrates that the value of the displacement transmissibility (dB) in the frequency range from zero to the first resonant peak could be controlled to less than zero and the displacement transmissibility is always below zero after the first resonant peak for larger \(n\) or \(l\) and smaller \(\theta\). Therefore, it can be concluded that the isolation effectiveness in a broad frequency band of the proposed isolation platform with SLSs can be improved by adjusting the structural parameters \(n\), \(\theta\), and \(l\) easily.