## Composition of the CLNVD and Solution of Its Amplification Coefficient
### The CLNVD's Displacement Amplification Coefficient and Energy Dissipation Coefficient
#### 2.2.2 The CLNVD's Effective Displacement Factor  
The derivation of the effective displacement factor \(\eta_{di}\) requires transforming the CLNVD's simplified mechanical model into the Kelvin model, which is displayed in Figure 4a. \(k\) is the equivalent stiffness of the lower intermediate column, the upper intermediate column, and the lever in series, which is calculated by Equation (11). In Equation (11), \(k_{le}\) is calculated by Equation (12), and \(k_{mc-up}\) is calculated by Equation (13). Equation (11) can refer to reference [18]. \(k_{le}\) is the stiffness of the lever amplification device, which is illustrated in Figure 4b. \(k_{mc-up}\) is the stiffness of the upper middle column after it is connected with the lever (See Figure 4c). \(k_{mc-low}\) is the stiffness of the lower middle column after it is connected with the lever (See Figure 4d). \(i_{mc}\) and \(i_{vmc}\) are the intermediate column's flexural line stiffness and shear line stiffness, respectively. \(i_{le}\) is the lever arm's bending line stiffness; \(l_{b}\) is the brace length; \(A_{b}\) is the brace's cross-sectional area; and \(\theta\) is the brace's horizontal included angle. \[k=\frac{k_{le}k_{mc-up}}{2k_{le}+k_{mc-up}} \tag{11}\]  
\[k_{le}=\frac{1}{\frac{(l_{1}+l_{2})^{2}\chi^{2}}{3l_{le}(\chi+1)^{2}}+\frac{ (\chi+1)^{2}l_{b}}{2 EA_{b}\cos^{2}\theta}} \tag{12}\]\[k_{mc-up}=\frac{1}{\frac{\chi^{2}[\beta f_{1}+(1-\beta)f_{2}]^{2}}{2f_{3}b}+\frac{ \chi^{2}(12t_{mc}+t_{mc}h^{2})}{12t_{mc}}} \tag{13}\]  
In Equation (5), \(f_{1}=(1+\frac{\beta}{\rho})[\beta(2-3\beta)-2(1+\frac{1}{\rho})(1-\beta)(3\beta -1)]\), \(f_{2}=[1+(1-\beta)/\rho]\)  
\([(1-\beta)(3\beta-1)-2\beta(1+1/\rho)(2-3\beta)]\), \(f_{3}=2-8(1+1/\rho)^{2}\). In order to achieve the transformation between the mechanical model of CLNVD and the Kelvin model, equal force, equal deformation, and equal energy dissipation are utilized here. The Kelvin model's damping coefficient is \(c^{\prime}_{a}\), and the spring stiffness is \(k^{\prime}_{a}\), which is shown in Figure 4a. Assuming the sinusoidal deformation generated by CLNVD, according to the definition of \(u_{ao}\), we obtain \(u_{ao}=\gamma_{i}U_{i}\). According to equal force and equal deformation, \(u_{d}(t)\), \(\dot{u}_{d}(t)\), and \(F(t)\) of VD are revealed as follows:  
\[\left\{ \begin{array}{l}u_{d}(t)=u_{ao}\sin(\omega t)-[k^{\prime}_{a}u_{ao} \sin(\omega t)+c^{\prime}_{a}\omega u_{ao}\cos(\omega t)]/(\gamma_{i}^{2}k)\\ \dot{u}_{d}(t)=\omega u_{ao}\cos(\omega t)-[k^{\prime}_{a}\omega u_{ao}\cos (\omega t)-c^{\prime}_{a}\omega^{2}u_{ao}\sin(\omega t)]/(\gamma_{i}^{2}k)\\ F(t)=ci_{d}(t)+k_{N}u_{d}(t)=[k^{\prime}_{a}u_{ao}\sin(\omega t)+c^{\prime}_{a }\omega u_{ao}\cos(\omega t)]/\gamma_{i}^{2}\end{array} \right. \tag{14}\]  
Equation (14) is utilized to solve the stiffness \(k^{\prime}_{a}\) and damping coefficients \(c^{\prime}_{a}\) in the Kelvin model. Details are as follows:  
\[\left\{ \begin{array}{l}k^{\prime}_{a}=\gamma_{i}^{2}k\Big{[}(k_{N}/k)^{2}+ k_{N}/k+(c\omega/k)^{2}\Big{]}/\left[(1+k_{N}/k)^{2}+(c\omega/k)^{2}\right]\\ c^{\prime}_{a}=\gamma_{i}^{2}c^{\prime}\Big{[}(1+k_{N}/k)^{2}+(c\omega/k)^{2} \Big{]}\end{array} \right. \tag{15}\]  
Figure 4: Schematic diagram of the CLNVD’s mechanical model and equivalent stiffness: (**a**) simplified mechanical model and Kelvin model; (**b**) \(k_{le}\); (**c**) \(k_{mc-up}\); (**d**) \(k_{mc-low}\). The loss stiffness of VD is defined as \(k_{d}^{\prime\prime}=c\omega\). \(\eta_{d}\) is solved based on equal energy consumption. Details are as follows:  
\[\eta_{d}=u_{do}/u_{ao}=1/\left[\left(1+k_{N}/k\right)^{2}+\left(k_{d}^{\prime \prime}/k\right)^{2}\right]^{0.5} \tag{16}\]  
When damping index \(\alpha=0\), \(k_{d}^{\prime\prime}=c/u_{do}\), \(k_{a}^{\prime}=\gamma_{i}^{2}k(k_{N}/k+k_{d}^{\prime\prime}/k)/(1+k_{N}/k+k_{ d}^{\prime\prime}/k)\), \(c_{a}^{\prime}=f_{d}\ c\), and \(\eta_{d}=1/(1+k_{N}/k+k_{d}^{\prime\prime}/k)\).