## 3 Structural Design and Analysis of QZS Vibration Isolation Systems  
### Negative Stiffness Element Design  
The negative stiffness element uses a fixed-end compressed buckling rectangular LS with a stiffness coefficient of k\({}_{1}\). The positive stiffness element uses a helical spring with a stiffness coefficient of k\({}_{2}\). As shown in Figure 6, these two components form a QZS system. After the complete assembly of the DCM, the weight of the periphery frame of the DCM compresses the helical spring, bringing the entire vibration isolation system to a balanced position where the total stiffness of the isolation system is approximately zero. Since the DCM contains precision components, such as voice coil motors and laser interferometer probes, the material used for making the LSs needs to have low magnetic permeability. ANSI301 is suitable as a material for LSs because, compared to other grades of stainless steel, ANSI301 has better strength and lower magnetic permeability, making it ideal for manufacturing elastic components in precision mechanical applications. In addition, DCMs have a strict volume limitation for working scenarios. Buckling leaf springs have a natural advantage in this regard and are better suited for this application than other types of negative stiffness structures. The rectangular LS plays a crucial role in the QZS system, serving as a key component in the design of the QZS system. Based on the specifications of the vibration isolation system, the length, width, and thickness of the LS are used as design parameters. Considering the practical aspects of manufacturing the LS, the thickness \(h\) of the LS is taken as a constant, while the length \(l_{0}\) or the width \(b\) of the LS are treated as variables for design and optimization. The QZS system, as shown in Figure 7, features an LS that bends into an arch shape under the action of compressive load \(F\). When subjected to external vibrations, the movable end of the LS undergoes a slight displacement \(\delta_{1}\) in the vertical direction. The elastic potential energy of QZS during the dynamic process satisfies the following equation:  
\[N\cdot\frac{\partial U_{a}}{\partial\delta_{1}}+\frac{U_{b}}{\partial\delta_{ 1}}=N\cdot(\frac{U_{c}}{\partial u_{x}}+\frac{U_{d}}{u_{x}})\cdot\sin(\tan^{-1 }(\frac{\delta_{1}}{l_{0}-u_{x}})) \tag{3}\]  
Figure 6: QZS vibration isolation system. In the equation, \(U_{a}\) represents the elastic potential energy generated by the deformation of the compliant LS due to the amplitude \(\delta_{1}\). \(U_{b}\) represents the elastic potential energy generated by the axial tension and compression of the positive stiffness spring. \(U_{c}\) represents the elastic potential energy generated by the compression of the compliant LS from its planar state (purple curve in Figure 7) to its buckled state (blue curve in Figure 7). \(U_{d}\) represents the elastic potential energy generated by the slight compression of the LS during its buckling process. \(N\) is the number of LSs in the QZS system. The terms in Equation (3) require detailed design parameters of the LS for calculation. Therefore, the following research will first establish a simplified force equilibrium equation for preliminary calculations, and the detailed derivation process of Equation (3) (the precise force equilibrium equation) will be presented in later sections. As shown in Figure 7, the fixed end of the LS is compressed towards the movable end by an amount \(u_{x}\). After the LS completes the buckling process, it stores elastic potential energies \(U_{c}\) and \(U_{d}\) in the horizontal direction. By differentiating these with respect to the compression \(u_{x}\), the compressive load \(F\) in the horizontal direction is obtained. When subjected to vibration excitation, the movable end of the LS displaces vertically by \(\delta_{1}\).