## Composition of the CLNVD and Solution of Its Amplification Coefficient  
### Composition and Features of the CLNVD  
Figure 1 illustrates the negative stiffness device's composition, the CLNVD's structure form, and the hysteresis curve. In the multi-story structure, CLNVD is installed between the upper beam and the lower beam, which is shown in Figure 0(a). The intermediate column's position can be selected as per the building space. Hence, CLNVD can be flexibly arranged in the span. The viscous damper (VD) and negative stiffness device (NSD) are linked in parallel to constitute an NVD.  
Figure 1: Intermediate column–lever negative stiffness viscous damper (CLNVD): (**a**) CLNVD; (**b**) NSD’s negative stiffness force; (**c**) hysteresis curves of NSD and NVD.  
When the inter-story deformation occurs, the beam drives the intermediate column to move, and the intermediate column moves the upper node of the lever arm through the connecting rod. The lever arm rotates through the node of the lever, causing significant deformation of the viscous damper and negative stiffness device. Due to the preloading effect of the internal spring, the deformation of NSD further increases, thereby increasing the deformation of VD. When the structure undergoes inter-story deformation, the node connecting the upper intermediate column and beam will rotate, causing the intermediate column to rotate. This means that the lower node of the upper intermediate column has undergone a rotational displacement, which is different from the structural inter-story deformation. When the intermediate column is located in the middle of the span, this rotational displacement is greater than the inter-story deformation of the structure. The lever arm causes significant deformation of NVD by rotating the node. When NSD deforms, the preload of the preload spring is released and acts on the piston rod of NSD through the convex surface, with the direction of this force being the same as the direction of the piston rod's movement. Therefore, when the NSD moves, the preloading spring will increase the displacement of the NSD. The viscous damper undergoes significant deformation under the combined action of the lever, lower intermediate column, and NSD, thereby resulting in significant energy dissipation to significantly control the seismic response of the structure.  
In Figure 1b, the convex surface shape of the piston rod is \(f(x)\); the stiffness and displacement of the preloaded spring are \(k_{SP}\) and \(\triangle_{SP}\), respectively; the number of springs is \(N\). Because the friction generated by the pulley rolling on the convex surface is much smaller than the viscous force generated by the viscous element VD, and in order to simplify the analysis, the friction force is ignored in this paper. The NSD's negative stiffness force \(f(x)\) is calculated through Equation (1). The specific derivation process can be found in reference [52].  
\[F(x)=N\frac{f^{\prime}(x)}{\left[1+f^{\prime}(x)\right]^{2}}k_{SP}[\Delta_{SP }+f(x)-f(0)] \tag{1}\]  
When NVD generates sinusoidal deformation, NVD and NSD's hysteresis curves are presented in Figure 1c. This sinusoidal deformation has an amplitude of 100 mm and a circular frequency of 1 rad/s. The spring stiffness is 1.2 kN/mm, and the preload displacement is 60 mm; the shape of the convex surface is \(f(x)=50\text{cos}(2\pi/600x)-7\text{cos}(2\pi/300x)\); the VD's damping coefficient \(c=2\text{ kN}/(\text{mm}/\text{s})\). In NSD, there are a total of 12 preload springs acting on the convex surface. The simplified curve in Figure 1c represents the hysteresis curve with a negative stiffness of \(-2\text{ kN}/\text{mm}\) and a damping coefficient of \(c=2\text{ kN}/(\text{mm}/\text{s})\). According to Figure 1c, the simplified curve is in good accordance with the theoretical curve. Therefore, NVD can be composed of linear negative stiffness elements and viscous elements in parallel.