Based on the heat diffusion equation, Maxwell’s equations, and translational and rotational dynamic equations, we establish and theoretically validate an electromagnetic-thermal-mechanical coupling model to analyze the levitation performance during normal operation and the nonlinear dynamic behavior under disturbance for 3D maglev systems composed of a six-degree-of-freedom bulk superconductor (SC) and a Halbach-type guideway of permanent magnets (PMs) with different magnetization strategies and different types of disturbances, as well as the change rules of magnetic force and torque during translational or rotational cycle movement. In order to ensure the system security, we propose a generalized electromagnetic restoring force model to theoretically analyze the stability of the SC moving along the directions of various degrees of freedom. The results show that after being disturbed, the SC vibrates along the direction of each degree of freedom, and the vibration center, i.e. equilibrium position, will drift along each vibration direction. With time increasing, the equilibrium position will appear periodically on both sides of the working position. Compared to zero-field cooling magnetization, field cooling magnetization enables the SC to trap more flux in its interior to alleviate the drift phenomenon and reduce the energy loss. This advantage can be further enhanced by adding an extra step of preloading treatment. For the lateral motion, the system has one stable focus point and two unstable saddle points. Whether the system at these saddle points is stable depends on the direction of disturbance-induced velocity. For the rotational motion, the system has only one stable focus point, which means that regardless of the type of disturbance, the SC will finally come back to its stable equilibrium position. Besides, the stability is related to the axis around which the SC rotates, and rotating around the longitudinal axis is more likely to generate larger magnetic force, torque and local temperature rise. Either field cooling magnetization or preloading treatment can effectively improve system stability.

中文翻译：

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六自由度 3D 超导磁悬浮的非线性动力学：对磁化策略和扰动类型的依赖

基于热扩散方程、麦克斯韦方程以及平移和旋转动力学方程，我们建立并理论上验证了电磁-热-机械耦合模型，以分析由 3D 磁悬浮系统组成的正常运行期间的悬浮性能以及扰动下的非线性动态行为研究了不同磁化策略和不同扰动类型的六自由度块体超导体（SC）和Halbach型永磁体（PM）导轨的磁力和扭矩在平动或运动过程中的变化规律。旋转循环运动。为了保证系统安全，我们提出了广义的电磁恢复力模型，从理论上分析了SC沿不同自由度方向运动的稳定性。结果表明，SC受到扰动后，沿各自由度方向振动，振动中心即平衡位置会沿各振动方向发生漂移。随着时间的增加，平衡位置会周期性地出现在工作位置的两侧。与零场冷却磁化相比，场冷却磁化使SC能够在其内部捕获更多磁通，从而减轻漂移现象并减少能量损失。通过添加额外的预加载处理步骤可以进一步增强这一优势。对于横向运动，系统具有1个稳定焦点和2个不稳定鞍点。这些鞍点处的系统是否稳定取决于扰动引起的速度的方向。对于旋转运动，系统只有一个稳定的焦点，这意味着无论扰动类型如何，SC最终都会回到其稳定的平衡位置。此外，稳定性与SC绕其旋转的轴线有关，绕纵轴旋转更容易产生较大的磁力、扭矩和局部温升。无论是场冷磁化还是预加载处理都可以有效提高系统的稳定性。