## 1 Introduction  
The generation of harmful vibration is inevitable, which reduces the performance of machines used in engineering [1; 2; 3]. Harmful vibration during engineering not only affects the use and working efficiency of precision instruments but also leads to precision instruments being damaged [4]. Therefore, it is particularly important to reduce harmful vibration in fields such as machining, transportation, and aerospace [4; 5]. One simple and effective way to reduce harmful vibrations is to use isolation devices [6]. These devices can realize vibration isolation in a wide range of frequencies, but most traditional isolators are linear, as they play a role in vibration isolation only when the excitation frequency is \(\sqrt{2}\) times higher than the natural frequency of the system [5; 7]. Therefore, if we want to achieve the low-frequency isolation of linear isolators, it is necessary to reduce the stiffness of their elastic elements. Selecting a lower system stiffness means that it will cause larger static displacement; however, the static bearing capacity of the system is insufficient. In terms of the deficiency of linear vibration isolation, vibration isolators using nonlinear damping or nonlinear stiffness designs have been of concern and, therefore, studied extensively in recent years. The introduction of nonlinear damping has been found to improve vibration isolation performance [8; 9], and the characteristic of nonlinear stiffness can broaden the frequency band of vibration isolation [10; 11; 12]. Both simulation and experiments show that the aforementioned nonlinear design methods can achieve better vibration isolation effects and have a high bearing capacity in the corresponding frequency band. Even so, the nonlinearity of this kind of special design needs higher application requirements and is hard to control, evaluate, and realize [2; 13; 14; 15]. Based on recent research investigating the contradiction between low stiffness and low static load in vibration isolation theory, we aim to analyze and design the isolator in a manner that reduces stiffness and increases its bearing capacity, thus improving the vibration isolation performance of the isolator. Silicone gel material, which is widely used in sports, medicine, and other fields, has excellent impact resistance and is a very promising buffer material for reducing shock and vibration [16; 17; 18; 19]. Based on its unique soft elastic properties, the resonance frequency of the SGC can reach the low-frequency region, meeting the requirements of the low-frequency vibration isolation of the isolator. The post-buckling behavior of the beam-column structure is part of the issue of supercritical deformation induced by axial force, and it has been studied by many scholars [20; 21; 22; 23]. When the beam-column structure is subjected to axial pressure and exceeds the critical pressure value, the buckling deformation of the structure will occur, and the post-buckling behavior of the super large deformation will occur if the pressure continues to increase [24]. Therefore, the design of the elastic elements of vibration isolators based on supercritical post-buckling deformation can ensure low stiffness while having the capability of initial static load, which greatly reduces the static displacement of the system [22; 25; 26]. By introducing a new silicone gel material to make the vibration isolation frequency reach the low-frequency region and using the elastic element design of post-buckling to improve the defect of insufficient static load, a new solution for low-frequency vibration isolation can be achieved. In this paper, we study the following aspects of the vibration isolation performance of the new vibration isolator. Firstly, based on two kinds of elliptic integral functions, the stiffness characteristics of the post-buckling spring are analyzed. Secondly, the dynamic models of the SGC vibration isolation systems of the post-buckling spring are established, and the approximate analytical solution is obtained using HBM.