## 3 Negative-Stiffness Isolation System and Isolation Principle  
The combined isolation system is composed of ordinary lead rubber bearings, belonging to the class of elastomeric bearings [13], and two pre-compression variable-stiffness isolation devices in parallel, and the parallel diagram is shown in Figure 13.  
The mechanical model of the negative-stiffness isolation system can be obtained by the parallel combination of the mechanical characteristics of the lead rubber bearing and the negative-stiffness device [12], as shown in Figure 14. As can be seen from Figure 14, the isolation system using the negative-stiffness stage of the variable-stiffness device in parallel with the lead rubber bearing cannot only amplify the displacement response of the isolation layer; it can also effectively reduce the seismic response of the superstructure. Therefore, the negative-stiffness part of the isolation composite system is additionally damped to achieve the purpose of reducing the displacement response of the isolation layer and the seismic response of the superstructure at the same time.  
Figure 13: Schematic diagram of variable-stiffness vibration isolation device.  
According to [PERSON] theory [14; 15], the horizontal stiffness of the bilinear model of lead rubber bearing is as follows:  
\[K_{2}=\frac{p^{2}}{2K_{r}q\tan(\frac{qh}{2})-ph} \tag{16}\]  
where \(q=\sqrt{p/K_{r}(1+p/K_{r})}\), \(p\) is compression load, \(h\) is the total thickness of rubber layer and sandwich steel plate, and \(K_{r}\) is the effective bending stiffness. The initial stiffness, \(K_{1}\) is 11 times that of the horizontal stiffness, \(K_{2}\), so the composite stiffness of the negative-stiffness isolation system is as follows [16]:  
Stiffness of the first stage:  
\[\begin{array}{ll}K_{1}+K_{NSD}&=\frac{11p^{2}}{2K_{d}q\tan(\frac{qh}{2})-ph}+ kf^{\prime}(x)\left[\frac{f^{\prime}(x)}{1+f^{\prime 2}(x)}-\mu\frac{1}{1+f^{ \prime 2}(x)}\right]\\ &+k[\Delta l+f(x)-f(x_{0})]\left[\frac{f^{\prime\prime}(x)[1+f^{\prime 2}(x)]-2f^{ \prime 2}(x)}{\left[1+f^{\prime 2}(x)\right]^{2}}+\mu\frac{2f^{\prime}(x)}{ \left[1+f^{\prime 2}(x)\right]^{2}}\right]\end{array} \tag{17}\]  
Stiffness of the second stage:  
\[\begin{array}{ll}K_{2}+K_{NSD}&=\frac{p^{2}}{2K_{d}q\tan(\frac{qh}{2})-ph}+ kf^{\prime}(x)\left[\frac{f^{\prime}(x)}{1+f^{\prime 2}(x)}-\mu\frac{1}{1+f^{ \prime 2}(x)}\right]\\ &+k[\Delta l+f(x)-f(x_{0})]\left[\frac{f^{\prime\prime}(x)[1+f^{\prime 2}(x)]-2 f^{\prime 2}(x)}{\left[1+f^{\prime 2}(x)\right]^{2}}+\mu\frac{2f^{\prime}(x)}{ \left[1+f^{\prime 2}(x)\right]^{2}}\right]\end{array} \tag{18}\]  
Considering the additional viscous damper, the dynamic equation of the isolated structure under the combined action of negative stiffness and damping is as follows:  
\[m\ddot{u}+(C_{1}+C_{h})\dot{u}+(k+K_{NSD})u=-m\ddot{u}_{g} \tag{19}\]  
where \(m\) is the mass of the isolated structure, \(u\) is the displacement of the isolation layer, \(C_{1}\) is the damping coefficient of lead rubber bearing, \(C_{h}\) is the damping coefficient of negative-stiffness device; \(k\) is the stiffness of lead rubber bearing, and \(K_{NSD}\) is the stiffness of the negative-stiffness device. The natural frequency of the traditional lead rubber bearing isolation structure is \(\omega_{0}=\sqrt{\frac{k}{m}}\), and the damping ratio of isolation structure is \(\xi_{0}=\frac{C_{1}}{2\pi u_{0}}\). The natural circular frequency of the structure with negative stiffness is \(\omega=\sqrt{\frac{k+K_{NSD}}{m}}=\omega_{0}\sqrt{1+\frac{K_{NSD}}{k}}\). When considering the damping effect of the device, the damping ratio of the isolation structure is as follows [16]:  
\[\xi=\frac{C_{1}+C_{h}}{2\pi\omega}=\frac{\xi_{0}(1+\frac{C_{h}}{C_{1}})}{\sqrt {1+\frac{K_{NSD}}{k}}} \tag{20}\]  
Figure 14: Mechanical model of negative-stiffness isolation system.  
According to the above equation, under the action of the damping negative stiffness, the damping structure self-vibration circle frequency decreases, and the structural damping ratio increases.