## 1 Introduction
For example, [PERSON] and [PERSON] [16] proposed an analytical model to calculate the magnetic force between two three-dimensional rectangular permanent magnets based on the surface charge model. [PERSON] et al. [17] designed an HSLD stiffness isolator using a negative-stiffness magnetic spring, and the magnetic force between the inner and the outer magnets was then calculated based on the Amperian current model. Magnetic springs have the outstanding advantages of compact design, efficient space utilization, no mechanical friction, and fewer vibration transmission paths. In particular, the stiffness characteristics can be adjusted by varying the distance or orientation of the magnets. Therefore, magnetic spring-based QZS vibration isolators find promising vibration control applications across numerous fields [18; 19]. More importantly, the loading capacity of a QZS vibration isolator strongly depends on the static positive stiffness. In most existing works, small payloads are always considered. However, there are still many heavy-load applications in practice, such as vehicle suspensions [20] and offshore platforms [21]. In these cases, a large-amplitude negative stiffness is always needed to match the large positive stiffness in order to generate the QZS. However, for existing single-layer magnetic rings, the negative stiffness near the equilibrium position is not always enough for heavy payloads. Therefore, it is very important to study novel magnetic structures with large-amplitude negative stiffness. The Halbach array has been proposed to greatly increase the magnetic force in a limited space, which is a special arrangement of permanent magnets that enhances the magnetic field on one side of the array [22; 23]. In the literature, there are two main types of Halbach arrays, namely flat Halbach arrays [24; 25] and Halbach cylinders [26]. In particular, Halbach cylinders are more suitable for compact design than flat Halbach arrays. The Halbach array offers a prior advantage over conventional layouts of magnets in terms of its concentrated magnetic-field intensity. Therefore, it is promising to introduce Halbach cylinders for designing novel magnetic springs. To the best of our knowledge, the application of Halbach arrays in QZS vibration isolators has not yet been reported. In particular, there are three key questions to be answered. (1) Can single-layer magnetic springs definitely generate negative stiffness or not? Can Halbach array-based magnetic springs definitely generate large-amplitude negative stiffness or not? (2) What kinds of Halbach array-based magnetic springs can generate large-amplitude negative stiffness? How can we select the optimal structural configuration? (3) What are the effects of geometric parameters on negative stiffness metrics of Halbach array-based magnetic springs? How can we determine the geometric parameters? To date, existing studies have seldom been carried out on Halbach array-based magnetic springs. Therefore, the motivation of this paper is to reveal nonlinear stiffness characteristics of Halbach array-based magnetic springs and investigate new magnetic springs with large-amplitude negative stiffness. Under the above-mentioned background, the novelty of the paper is to present the Halbach-cylinder magnetic spring (HCMS) for heavy-load QZS by using Halbach cylinders. The HCMS is composed of two coaxial Halbach cylinders, where the outer cylinder is fixed and the inner cylinder can move freely. Key metrics of the HCMS include the amplitude, the uniformity, and the displacement range of negative stiffness near the equilibrium position. The main challenges include deriving an analytical formula for nonlinear stiffness and designing proper structural configurations. In summary, the main contributions of this work can be outlined as follows:  
(1) The idea of Halbach array-based magnetic springs is introduced, so it is possible to generate large-amplitude negative stiffness by using magnetic structures.