Magnetic stiffness is as important as vertical and lateral forces to achieve stable levitation in high-temperature superconductor (HTS) levitation systems. To date, research on magnetic stiffness has mainly focused on a few aspects, but the understanding of its comprehensive characteristics has not been included. This study is focused on the quantitative properties of the physical and geometrical parameters that affect the vertical stiffness of HTS levitation systems using a numerical approach when a permanent magnet (PM) above the HTS bulk moves vertically under both zero-field cooling and field cooling (FC) conditions. The results show that the magnetic field of the PM, the critical current density (Jc), and the PM to HTS area ratio (α) significantly affect the vertical stiffness, while the PM to HTS thickness ratio has little effect. In addition, some unexpected phenomena were discovered. At some large vertical distances, the vertical stiffness first decreased and then increased as the external magnetic field increased under FC. The phenomenon of sign switching for the vertical stiffness appears to be related to several parameters, such as the external magnetic field, Jc, and α. At most vertical distances (except positions at which the saturation characteristic of vertical stiffness appears), the vertical stiffness increases with α. These findings are different from the results of the vertical force dependence with the same parameters.

中文翻译：

---

当永磁体在高温超导体上方垂直移动时，物理和几何参数对垂直磁刚度的影响

要在高温超导体 (HTS) 悬浮系统中实现稳定悬浮，磁刚度与垂直和横向力一样重要。迄今为止，对磁刚度的研究主要集中在几个方面，但对其综合特性的理解尚未包括在内。当 HTS 体上方的永磁体 (PM) 在零场冷却和场冷却下垂直移动时，这项研究的重点是使用数值方法影响 HTS 悬浮系统垂直刚度的物理和几何参数的定量特性。 FC) 条件。结果表明，PM的磁场、临界电流密度（Jc）和PM与HTS的面积比（α）显着影响垂直刚度，而PM与HTS的厚度比影响不大。此外，还发现了一些意想不到的现象。在一些较大的垂直距离处，垂直刚度随着外部磁场的增加先降低然后增加。垂直刚度的符号转换现象似乎与几个参数有关，例如外部磁场、Jc 和 α。在大多数垂直距离处（除了出现垂直刚度饱和特性的位置），垂直刚度随 α 增加。这些发现与具有相同参数的垂直力依赖性的结果不同。垂直刚度的符号转换现象似乎与几个参数有关，例如外部磁场、Jc 和 α。在大多数垂直距离处（除了出现垂直刚度饱和特性的位置），垂直刚度随 α 增加。这些发现与具有相同参数的垂直力依赖性的结果不同。垂直刚度的符号转换现象似乎与几个参数有关，例如外部磁场、Jc 和 α。在大多数垂直距离处（除了出现垂直刚度饱和特性的位置），垂直刚度随 α 增加。这些发现与具有相同参数的垂直力依赖性的结果不同。