## 3 Structural Design and Analysis of QZS Vibration Isolation Systems
### Negative Stiffness Element Design
At this point, the positive stiffness spring stores elastic potential energy \(U_{b}\) due to the change in length in the vertical direction, and the LS stores elastic potential energy \(U_{d}\) due to the displacement of the movable end in the vertical direction. Differentiating these with respect to \(\delta_{1}\) yields the elastic forces in the vertical direction, which are opposite in direction to the displacement \(\delta_{1}\). Furthermore, after the displacement of the movable end of the LS, the direction of the compressive load \(F\) of the LS inclines by an angle \(\theta\), which is the angle between the line connecting the movable end and the fixed end of the LS and the horizontal line. This causes a component of the compressive load \(F\) to act in the vertical direction, in the same direction as \(\delta_{1}\). Considering these conditions, under external vibration excitation, the system's response amplitude is \(\delta_{1}\). The forces exerted by the positive stiffness spring and the LS in the vertical direction (left side of Equation (3)) balance with the vertical component of the \(F\) from the buckled LS (right side of Equation (3)). Therefore, the force balance equation during the vibration process is obtained as shown in Equation (3). In Equation (3), the \(U_{c}\) term is not calculated using the deflection curve during the vibration process (orange curve in Figure 7) because the deformation of the LS during the vibration process is mainly in the vertical direction, and its compression potential energy remains almost unchanged. Hence, the error introduced by using the equilibrium position's deflection curve (blue curve in Figure 7) for the calculation is minimal. In the actual design process, all the dimensional parameters of the LS are required to establish the force balance equation. Therefore, at the initial design stage, the force balance equation needs to be simplified to avoid missing dimensional parameters. After completing the preliminary calculations using the simplified formula, the precise equation can be used for adjustments based on the available conditions. The \(F\) of the LS in the  
Figure 7: Elastic potential energy in the QZS system. horizontal direction is simplified to an unknown quantity \(P\). The elastic potential energy \(U_{a}\), due to the deformation of the LS in the vertical direction, is relatively small, and, since it cannot be calculated without the dimensional parameters of the LS, it can be temporarily ignored. Thus, Equation (3) simplifies to the following:  
\[k_{2}\cdot\delta_{1}=N\cdot P\cdot\sin(\tan^{-1}(\frac{\delta_{1}}{l_{0}-u_{x}})) \tag{4}\]  
The axial compression \(u_{x}\) is typically two orders of magnitude smaller than the initial length \(l_{0}\). Additionally, the derivatives of the sine and arctangent functions near the origin are approximately one. Therefore, the equation can be approximated as follows:  
\[k_{2}\cdot\delta_{1}=N\cdot P\cdot\frac{\delta_{1}}{l_{0}} \tag{5}\]  
In the operation of the QZS system, the negative stiffness element must always remain in a state of compressed energy storage, meaning it needs to have sufficient compression to prevent the free end displacement from fully extending and losing the negative stiffness effect during vibrations. The prerequisite for the LS to achieve a larger compression amount is that the \(F\) must be greater than the critical load \(P_{\sigma}\) for its first mode shape buckling, as shown in Figure 8. There is a nonlinear relationship between the axial compression amount and the compression load, primarily characterized by a stiffness softening effect occurring in the LS's axial compression once the compression load exceeds \(P_{\sigma}\). The calculation method for this will be detailed later. As shown in Figure 8d, the rectangular LS used in the quasi-zero stiffness system has two free edges, which are neither loaded nor constrained. Therefore, its deflection calculation can be equivalent to the deflection calculation of a beam.