## 3 Structural Design and Analysis of QZS Vibration Isolation Systems
### Potential Energy and Negative Stiffness Derivation  
In the previous theoretical derivation, all parameters necessary for calculating potential energy have been obtained. In the calculation of compression, the elastic potential energy \(U_{c}\) of the compliant LS was derived from Equation (13). Next, we proceed to calculate the elastic potential energies \(U_{a}\), \(U_{b}\), and \(U_{d}\). For \(U_{a}\), we can similarly use the approach in Equation (7): calculating the elastic potential energy of the one end fixed and one end with small displacement (the orange deflection curve in Figure 7) and subtracting the elastic potential energy due to compression alone (the blue deflection curve in Figure 7). For the calculation of the deflection curve with one end fixed and one end with small displacement after compression buckling (the orange deflection curve in Figure 7), the superposition method can be used by superimposing the buckled deflection curve with both ends fixed and one end eccentric on the buckled deflection curve with both ends fixed. As shown in Figure 10, the moment distribution of a beam fixed at both ends and eccentric at one end is as follows:  
\[M=F(\frac{\delta_{1}}{\gamma}-w_{2}) \tag{15}\]  
Figure 9: Theoretical calculation of compression displacement compared with finite element simulation. \(E\) of the LS is 193 GPa, \(b\) is 80 mm, and \(l_{0}\) is 70 mm. where \(w_{2}\) is the deflection of the beam. The differential equation for the deflection curve can be obtained from the moment:  
\[EIw^{\prime\prime}_{2}=-M=F(\frac{\delta_{1}}{2}-w_{2}) \tag{16}\]  
Using the notation \(k^{2}=F/EI\), Equation (16) becomes the following:  
\[w^{\prime\prime}_{2}+k^{2}w_{2}=k^{2}\frac{\delta_{1}}{2} \tag{17}\]  
The homogeneous solution \(w_{H}\) and particular solution \(w_{H}\) of this equation are as follows:  
\[\left\{ \begin{array}{l}w_{H}=C_{1}\sin kx+C_{2}\cos kx\\ w_{p}=\frac{\delta_{1}}{2}\end{array} \right. \tag{18}\]  
Using the boundary conditions, we can obtain the following:  
\[\left\{ \begin{array}{l}w_{2}(0)=0\\ \mathrm{w}_{2}(1_{0}-\mathrm{u}_{x})=\delta_{1}\\ w^{\prime}_{2}(0)=\ \mathrm{w}\ t_{2}(1_{0}-\mathrm{u}_{x})=0\end{array} \right. \tag{19}\]  
Substituting into Equation (18) gives the following:  
\[\left\{ \begin{array}{l}C_{1}=0\\ C_{2}=-\frac{\delta_{1}}{2}\\ k=\frac{\pi}{0-\mathrm{u}_{x}}\end{array} \right. \tag{20}\]  
The deflection equation is as follows:  
\[w_{2}=\frac{\delta_{1}}{2}(1-\cos\frac{\pi x}{l_{0}-u_{x}}) \tag{21}\]  
As shown in Figure 11, using the superposition method to simply add \(w_{1}\) and \(w_{2}\), the deflection curve differs from the finite element simulation results primarily in that the maximum deflection in the finite element simulation is slightly greater than the approximate value. Moreover, the error is larger on the side near the fixed end when \(\delta_{1}>0\), whereas the error is larger on the side near the movable end when \(\delta_{1}<0\). Since the deflection curve calculated using the superposition method already includes the deflection curve due to buckling alone (the blue deflection curve in Figure 7), the calculation of potential energy  
Figure 10: Buckling deflection curve of a beam with both ends fixed and one end eccentrically loaded. \(U_{a}\) can be simplified to the energy stored by the LS bending into the shape \(w_{2}\) under the moment, thereby omitting the calculation of elastic potential energy due to buckling. \[U_{a}=\frac{EI}{2}\int{(\frac{d^{2}w_{2}}{d\chi^{2}})}^{2}\ dx \tag{22}\] \[=\frac{EI\delta_{1}^{2}\pi^{4}}{16(l_{0}-u_{x})^{3}}\]  
It should be noted that the curve \(w_{2}\) undergoes compressive buckling in the horizontal direction; however, here, its curve \(w_{2}\) is calculated only to be superimposed on \(w_{1}\) in the vertical direction, so the subsequent derivation process only needs to calculate the elastic force of the curve \(w_{2}\) in the vertical direction. In addition, compression buckling is a large deformation behavior.