## 2 Negative-Stiffness Device and Its Mechanical Model
### Mechanical Model  
Set the initial spring length as \(l_{p}\), the spring preload as \(\Delta l\), the spring length after initial compression as \(l_{0}\), the spring stiffness as \(k\), and the friction coefficient of the track's surface as \(\mu\). In the initial state, the position of the roller is in the coordinate system shown in Figure 2a; the function value of the track surface is \(f(x_{0})\), and the spring length at any time is \(l\). With the horizontal displacement of the device, \(u\), the function value of the track surface where the roller is located in the coordinate system, as shown in Figure 2b, is \(f(x)\); the stress state of the corresponding roller is shown in Figure 3. At this time, the spring length is as follows:  
\[l=l_{0}+f(x)-f(x_{0}) \tag{1}\]  
Figure 1: Schematic diagram of damping-track-type variable-stiffness device. Figure 2: Motion state of negative-stiffness device. (**a**) Initial state of the device (**b**) Device state after horizontal displacement, u. The corresponding spring force, \(F\), and the rail surface reaction, \(N\), are calculated as follows:  
\[F=k(l_{p}-l) \tag{2}\]  
\[N=F\cos\theta \tag{3}\]  
where \(\theta\) is the angle between the tangent of the rail surface function at the roller position and the \(x\)-axis. At this time, the friction force in the tangent direction of the track surface where the roller is located is as follows:  
\[F_{f}=\mu F\cos\theta \tag{4}\]  
Decompose \(N\) and \(F_{f}\) to the horizontal direction, and it can be concluded that the force provided by the spring part of the track-type-damping negative-stiffness device opposite to the displacement direction, \(F_{NSDspring}\), is as follows:  
\[F_{NSDspring}=F\cos\theta\sin\theta-\mu F\cos^{2}\theta=k(l_{p}-l)\cos\theta \sin\theta-\mu F\cos^{2}\theta \tag{5}\]  
According to the conversion relationship between trigonometric functions, we have the following:  
\[f^{\prime}(x)=\tan\theta\]  
\[\sin\theta\cos\theta=\frac{\tan\theta}{1+\tan^{2}\theta} \tag{6}\]  
\[l=l_{0}+f(x)-f(x_{0}) \tag{1}\]0  
It can be concluded that the formula for \(F_{NSDspring}\), as expressed by the orbital surface function, is as follows:  
\[l=l_{0}+f(x)-f(x_{0}) \tag{1}\]1  
Calculate the first derivative of the displacement \(x\) in Formula (6) to obtain the relationship between the negative stiffness of the device and the displacement:  
\[ \begin{array}{ll}K_{NSD}&=kf^{\prime}(x)\bigg{[}\frac{f^{\prime}(x)}{1+\left[ f^{\prime}(x)\right]^{2}}-\mu\frac{1}{1+\left[f^{\prime}(x)\right]^{2}} \bigg{]}\\ &+k[\Delta l+f(x)-f(x_{0})]\bigg{[}\frac{f^{\prime\prime}(x)[1+f^{\prime 2 }(x)]-2f^{\prime 2}(x)}{\left[1+f^{\prime 2}(x)\right]^{2}}+\mu\frac{2f^{ \prime}(x)}{\left[1+f^{\prime 2}(x)\right]^{2}}\bigg{]}\end{array}  \tag{8}\]  
Let the damping coefficient of the velocity dependent viscous damper be \(C\), and we can see that the initial stiffness takes 100 times of the damping coefficient \(C\)[11]. Under the action of ground motion, with the horizontal displacement of the track plate, the relative position of the roller on the track surface changes, and this will produce a vertical velocity,  
Figure 3: Stress-state analysis of roller. i.e., the axial velocity of the viscous damper, which is set as \(v\). Then the relationship between \(v\) and track surface function \(f(x)\) is calculated as follows:  
\[v=\frac{df(x)}{dt}=\frac{df(x)}{dx}\frac{dx}{dt}=f^{\prime}(x)\cdot\dot{x} \tag{9}\]  
Then the damping force generated by the viscous damper is decomposed to the horizontal direction as follows (the decomposition method is the same as the spring part):  
\[F_{NSDdamper} =c|v|^{\partial}\cdot\text{sgn}(\dot{x}) \tag{10}\] \[=c\cdot\text{sgn}(\dot{x})\cdot|f^{\prime}(x)\cdot\dot{x}|^{ \partial}\left[\frac{f^{\prime}(x)}{1+f^{\prime 2}(x)}-\text{sgn}(\dot{x})v\frac{1}{1+f^{ \prime 2}(x)}\right]\]  
In this paper, the commonly used \(\alpha=0.3\) is taken to discuss the mechanical properties of the device.