## 1 Introduction
At present, the field of QZS vibration isolation has produced many excellent research outcomes. For example, in terms of structural and material innovations, [PERSON] et al. studied a QZS system based on 3D-printed soft resin materials. Due to the damping properties of the resin material itself, this structure achieved superior vibration isolation performance and robustness to excitation level changes within the low-frequency range [27]. [PERSON] et al. proposed a compact one-dimensional QZS metamaterial composed of conical shell structural units featuring adjustable bandgaps. This structure offers advantages such as small size, light weight, compactness, and high reliability. The isolation starting frequency was achieved at 15.01 Hz, exhibiting good vibration isolation performance [28]. [PERSON] et al. introduced a QZS isolator with dual-arc-shaped flexible beams. This structure can adapt to different vibration isolation requirements, achieving the desired isolation performance by merely modifying the assembly parameters. Additionally, the structure can accommodate load variations within 10% and provides excellent vibration isolation for excitations above 4 Hz [29]. [PERSON] et al. proposed a QZS system based on a conical origami structure. The research showed that this structure, compared to existing QZS isolators, has a broader isolation bandwidth, a lower starting frequency, and reduced transmissibility, demonstrating superior isolation performance, particularly at low frequencies [30]. The QZS vibration isolation system studied in this paper uses fixed-end buckling leaf springs as the negative stiffness element, which is structurally different from the aforementioned research results. Therefore, theoretical studies must refer to other relevant outcomes. Several studies are structurally similar to the QZS negative stiffness elements studied in this paper. For instance, [PERSON] et al. designed a new hyperbolic QZS structure with nonlinear stiffness, integrating a pair of cosine beams (negative stiffness elements) and a pair of arc beams (positive stiffness elements) for low-frequency vibration isolation. The research showed that the required positive, zero, or negative stiffness could be achieved by selecting appropriate geometric parameters [31]. [PERSON] et al. proposed using B-spline beams as negative stiffness elements, derived the B-spline beam configuration, and discussed the relationship between the beam's geometric parameters and mechanical properties. Finite element simulations and experiments proved the B-spline beam's negative stiffness structure to be repeatable and efficient in terms of energy absorption [32]. The QZS vibration isolation system studied in this paper uses fixed-end buckling leaf springs as the negative stiffness elements, which is structurally different from the aforementioned research results. As shown in Figure 2, most studies with structures similar to this one use the configuration shown in Figure 2a to achieve compression buckling. These studies have also made significant progress in the study of negative stiffness. For example, [PERSON] et al. theoretically predicted the negative stiffness of compressed buckling beams under large deformation conditions using the elliptical integral method. The experimental results showed that the negative stiffness values calculated using this method had high accuracy [33]. However, in this type of negative stiffness element, the compression amount is entirely dependent on the structural size parameters and is not an adjustable design variable. In this paper, the configurations shown in Figure 2b,c are used, where the compression amount is treated as an independent design variable that can be freely adjusted in applications. Therefore, the structure studied in this paper cannot directly use the elliptical integral method to solve the negative stiffness calculation problem. There are relatively few studies with negative stiffness elements with the same structure and boundary conditions as those in this paper. For example, [PERSON] et al.