## 3 Parameter Analysis and Optimization Design
### Parameter Analysis
#### 3.1.5. Impact of NSD's Negative Stiffness  
This section analyzes the variation in \(f_{d}\) and \(f_{E}\) with the changes in NSD's \(k_{N}\). As is shown in Figure 9(a), when \(k_{N}\) decreases, \(\eta_{d}\) gradually increases. Therefore, \(f_{d}\) and \(f_{E}\) gradually increase. From Figure 9(b), it is concluded that when \(k_{N}\) decreases, \(k_{d}^{\prime}\) decreases at first and then increases. Moreover, a smaller \(c\) will give CLNVD larger negative stiffness. Therefore, the size of \(k_{N}\) should be controlled to make CLNVD achieve a higher \(f_{d}\) while avoiding an influential decrease in the structure's overall stiffness. SAP2000 (Version 23.1.0) is used for numerical simulation. The results indicate that the theoretical calculation results and numerical simulation results of \(f_{d}\) are very consistent under different \(k_{N}\) conditions. The CLNVD can be guaranteed to achieve a high \(f_{d}\) by controlling the size association between \(k_{N}/k\) and \(k_{d}^{\prime\prime}/k\). The reference [54] requires that the ratio of the viscous damper's loss stiffness to the brace stiffness is less than \(1/3\). When \(f_{d}=1/(1+1/3^{1+\alpha})^{1-0.5\alpha}\gamma\), \(f_{d}=\gamma\), and \(f_{d}=2\gamma\), the size relationship between \(k_{N}/k\) and \(k_{d}^{\prime\prime}/k\) can be calculated from Equation (22) as follows:  
\[\begin{cases}k_{N}/k=\left[1+1/3^{1+\alpha  
The size of CLNVD's \(k_{a}^{\prime}\) can also be controlled by controlling the size association between \(k_{N}/k\) and \(k_{d}^{\prime\prime}/k\). When \(k_{a}^{\prime}=\gamma^{2}k_{N}\) or \(k_{a}^{\prime}=-\gamma^{2}k\), the size association between \(k_{N}/k\) and \(k_{d}^{\prime\prime}/k\) are showed in Equation (23).  
\[\begin{cases}(1+k_{N}/k)^{\alpha}(k_{N}/k)^{2}/(1-k_{N}/k)=(k_{d}^{\prime \prime}/k)^{1+\alpha}&,\,k_{a}^{\prime}=\gamma^{2}k_{N}\\ (1+k_{N}/k)^{\alpha}(1+2k_{N}/k)=-2(k_{d}^{\prime\prime}/k)^{1+\alpha}&,\,k_ {a}^{\prime}=-\gamma^{2}k\end{cases} \tag{23}\]  
Combining Equation (22) as well as Figure 10(c), we can see that when \(k_{N}/k\leq\left[1+1/3^{1+\alpha}-(k_{d}^{\prime\prime}/k)^{1+\alpha}\right]^ {1/(1+\alpha)}-1\), \(f_{d}\geq 1/(1+1/3^{1+\alpha})^{1-0.5\alpha}\gamma\). Combining Equation (23) and Figure 10(d), we can see that when \((1+k_{N}/k)^{\alpha}(k_{N}/k)^{2}/(1-k_{N}/k)\geq(k_{d}^{\prime\prime}/k)^{1+ \alpha}\), \(k_{a}^{\prime}\geq\gamma^{2}k_{N}\). When \((1+k_{N}/k)^{\alpha}(1+2k_{N}/k)\leq-2(k_{d}^{\prime\prime}/k)^{1+\alpha}\), \(k_{a}^{\prime}\leq-\gamma^{2}k\). Through Equation (19), it can be obtained that when \(k_{N}/k\geq-0.5\), \(k_{a}^{\prime}\geq-\gamma^{2}k\). We can see from Section 3.1.4 that when \(\left(k_{d}^{\prime\prime}-k_{N}\right)/k\leq 1\), \(f_{E}\) increases as the damping coefficient \(c\) increases. Thus, according to the above analysis, it is recommended that the target area in Figure 10(d) should be taken as the range of values for \(k_{N}/k\) and \(k_{d}^{\prime\prime}/k\). Within this target area, we can see the following: \(f_{d}\geq 1/(1+1/3^{1+\alpha})^{1-0.5\alpha}\gamma\gamma^{2}k_{N}\leq k_{a}^{\prime}\); \(f_{E}\) rises when increasing the damping coefficient \(c\).