## Composition of the CLNVD and Solution of Its Amplification Coefficient
### The CLNVD's Displacement Amplification Coefficient and Energy Dissipation Coefficient
#### 2.2.1 The CLNVD's Geometric Amplification Coefficient  
The force analysis of the \(i\)-th story in the structure shown in Figure 1a is conducted (See Figure 2a). \(F\) is the horizontal force of the \(i\)-th story; \(P_{1}\)-\(P_{3}\) is the column axial force of the \(i\) + 1 st story; \(M_{1}\)-\(M_{3}\) is the bending moment at the column bottom on the \(i\) + 1 st story; \(\triangle_{1}-\triangle_{3}\) is the cumulative axial deformation of the column from the first story to the \(i\) - 1 th story; and \(\theta\) is the corner angle at the column bottom on the \(i\)-th story. \(U_{i}\) is the inter-story displacement on the \(i\)-th story. \(u_{ao}\) is the displacement of VD on the \(i\)-th story when \(c\) = 0 and \(k_{N}\) = 0. The isolation body illustrated in Figure 2a has the following relationship:  
\[U_{i}=U_{Fi}+U_{Mi}+U_{\Delta i}+U_{\theta i} \tag{5}\]  
\[u_{ao}=u_{Fao}+u_{Mao}+u_{Bao}+u_{\Delta ao} \tag{6}\]  
In Equations (5) and (6), \(U_{Fi}\), \(U_{Mi}\), \(U_{\Delta i}\), and \(U_{\theta i}\) are the inter-story displacements under the action of \(F\), \(M\), \(\triangle\), and \(\theta\), respectively. \(u_{Fao}\), \(u_{Mao}\), \(u_{Aao}\), and \(u_{Bao}\) are displacements of VD under the action of \(F\), \(M\), \(\triangle\), and \(\theta\), which is shown from Figure 2b-e. They are defined as follows:  
\[\rho=i_{b}/i_{c},\ \beta=a\ /l,\ \chi=l_{2}/l_{1} \tag{7}\]  
\[\gamma_{F}=u_{Fao}/U_{Fi}\,\ \gamma_{M}=u_{Mao}/U_{Mi}\,\ \gamma_{\theta}=u_{ Bao}/U_{\theta i}\,\ \gamma_{\Delta}=u_{\Delta ao}/U_{\Delta i} \tag{8}\]  
In Equations (7) and (8), as shown in Figure 1, \(a\) is the distance between the intermediate column and the left column. \(l\) is the span length, and \(h\) is the floor height. \(l_{1}\) and \(l_{2}\) are the lengths of the lever arm at the upper and lower ends; \(i_{b}\) and \(i_{c}\) are the beam and column's bending line stiffness, respectively. \(\beta\) represents the intermediate column's position, \(\rho\) represents the stiffness ratio of the beam to column, and \(\chi\) represents the lever's amplification coefficient. The coefficients in Equation (8) are analyzed. \(\gamma_{F}\), \(\gamma_{M}\), \(\gamma_{\theta}\), and \(\gamma_{\Delta}\) is expressed as follows:  
\[\left\{ \begin{array}{l}\gamma_{F}=\chi-3\chi\big{[}6(\beta^{2}-\beta)+1 \big{]}/\big{[}2(3\rho+2)\big{]}\\ \gamma_{M}=\gamma_{\theta}=6\chi(\beta-\beta^{2})\\ \gamma_{\Delta}=\chi(6+2/\rho)(\beta^{2}-\beta)\end{array} \right. \tag{9}\]  
Equation (9) can be found in reference [15]. And the derivation of \(\gamma_{F}\) can be found in Appendix A. By combining Equations (2), (5), (6), and (8), we obtain the geometric amplification coefficient as follows:  
\[\gamma_{i}=\frac{u_{ao}}{U_{i}}=\frac{\gamma_{F}U_{Fi}+\gamma_{M}U_{Mi}+\gamma _{\theta}U_{\theta i}+\gamma_{\Delta}U_{\Delta i}}{U_{Fi}+U_{Mi}+U_{\theta i}+U_ {\Delta i}} \tag{10}\]According to Equation (10), \(\gamma_{i}\) is related to \(\gamma_{F}\),\(\gamma_{M}\),\(\gamma_{\theta}\), and \(\gamma_{\Delta}\). Figure 3 reveals the variation pattern of \(\gamma_{F}\), \(\gamma_{M}\), \(\gamma_{\theta}\), and \(\gamma_{\Delta}\) with \(\rho\) and \(\beta\) when \(\chi=4\). As \(\beta\) increases, \(\gamma_{F}\) increases at first, and then decreases, reaching its maximum value when \(\beta=0.5\). There are two commonalities in \(\gamma_{F}\) under different \(\rho\). \(\beta_{\rm A}=1/2-1/\sqrt{12}\), \(\beta_{\rm B}=1/2+1/\sqrt{12}\). When \(\beta_{\rm A}<\beta<\beta_{\rm B}\), \(\gamma_{F}\) decreases as \(\rho\) increases. As is shown in Figure 3b, we can see that just like the \(\gamma_{F}\), \(\gamma_{M}\) also reaches its maximum value when \(\beta=0.5\). From Figure 3c, it can be seen that when \(\beta\in(0,1)\), \(\gamma_{\Delta}\) decreases as \(\rho\) decreases, and all values are negative. Therefore, the cumulative axial deformation of the column reduces the geometric amplification coefficient \(\gamma_{i}\).