## 5 Conclusions  
The CLNVD consists of the viscous damper, the negative stiffness device, the lever, and the intermediate column, which has a low demand for stiffness of the connecting components and enhances the damper's displacement. In this paper, energy dissipation coefficient \(f_{E}\) and displacement amplification coefficient \(f_{d}\) are proposed to intensively assess the CLNVD's performance. The coefficient \(\gamma\) is introduced to describe the impact of the intermediate column's position on the CLNVD's amplification capability, and the factor \(\eta_{d}\) is introduced to describe the weakening of the equivalent stiffness on the CLNVD's  
Figure 19: Story shear forces under fortification earthquakes and rare earthquakes: (**a**) fortification earthquakes; (**b**) rare earthquakes. amplification capability. Moreover, the impacts, such as column's position, beam's stiffness, lever's amplification coefficient, negative stiffness, damping coefficient, and inter-story displacement on CLNVD's \(f_{d}\) and \(f_{E}\), are studied. On this basis, this paper proposes a damping design method to optimize the CLNVD's amplification capability. The main conclusions are as follows:  
(1) The CLNVD can employ \(f_{d}\) and \(f_{E}\) to, respectively, describe its displacement amplification capability and energy dissipation capability. The coefficient \(f_{d}\) is related to the geometric amplification coefficient \(\gamma\) as well as the effective displacement factor \(\eta_{d}\). (2) CLNVD's \(f_{d}\) and \(f_{E}\) will be maximized, and \(f_{E}\) will increase first and then decrease as the beam's stiffness increases when the column's position is \(\beta=0.5\). (3) The amplification capability of CLNVD will increase when increasing the lever's amplification coefficient \(\chi\). When the negative stiffness is 0, there exists an optimum lever amplification coefficient that maximizes CLNVD's \(f_{E}\). (4) There exists an optimal combination of damping coefficient c and index \(\alpha\) that maximizes CLNVD's \(f_{E}\). Under this optimal combination, c and \(\alpha\) need to satisfy \(k_{d}^{\prime\prime}/k=1\)\(+k_{N}/k\) or \(\eta_{d}=1/[2(1+k_{N}/k)\ ^{1+\alpha}]\ ^{1-0.5\alpha}\). (5) In contrast to the intermediate column damping system or the intermediate column-lever damper, the negative stiffness device enhances the displacement of CLNVD. In order to enable CLNVD to obtain greater displacement while avoiding providing too much negative stiffness, it is very necessary to make \(k_{N}/k\) and \(k_{d}^{\prime\prime}/k\) within the target area in Section 3.1.5. Within this area, \(f_{d}\geq 1/(1+1/3^{1+\alpha})^{1-0.5\alpha}\gamma\), \(\gamma^{2}k_{N}\leq k_{d}^{\prime}\), and \(f_{E}\) increases when increasing damping coefficient \(c\). (6) In order to enable CLNVD to have high energy dissipation capacity in both small and large displacements, the following methods can be utilized: to increase \(\chi\) and \(i_{b}\) simultaneously; to increase \(i_{b}\) and \(c\); or to increase \(\chi\) while decreasing \(c\). (7) In vibration-reduction design, applying the optimization design method in Section 3.2 can enable CLNVD to effectively control structure's seismic response. The CLNVD provides a damping ratio of 20.5% in Scheme 1 under fortification earthquakes. Under the circumstance of rare earthquakes and fortification earthquakes, the inter-story displacement has a 50% reduction effect. Meanwhile, this optimization design method can provide a reference for vibration-reduction design in other types of negative stiffness dampers with amplification function. Conceptualization, Q.Z. and W.P. ; software, Q.Z. and Z.L. ; validation, Q.Z., W.P., X.L. and Z.L. ; formal analysis, Q.Z. ; data curation, Q.Z. and Z.L. ; writing--original draft preparation, Q.Z. ; writing--review and editing, Q.Z. and X.L. ; visualization, X.L. ; project administration, W.P. ; funding acquisition, W.P. All authors have read and agreed to the published version of the manuscript.