Magnetic levitation by rotation described by a new type of Levitron

TL;DR

The paper investigates a dynamical Levitron where a rotating rotor magnet and a nearby floater magnet produce magnetic levitation with the floater's effective mass-to-magnetic-moment ratio $m/\mu_f$ modulated by the lateral offset $\delta_R$. By modeling the floater as a magnetic dipole and deriving the potential energy $U_f(z_f)$ including dipole-dipole interactions and gravity, the authors distinguish decoupled ($\delta_R \approx 0$) and coupled ($\delta_R \gtrsim \delta_R^c$) regimes and identify conditions for a local energy minimum that yields trapping. They find a finite critical offset $\delta_R^c$ around $0.3$–$0.4$ mm, enabling trapping (even with the floater at rest), and an upper bound $\delta_R^{\max}$ near $2.5$ mm beyond which trapping ceases; dynamical trapping with nonzero floater rotation requires a lower bound on $\delta_R$ that scales with the floater's rotation, with drag contributing ~20% deviations from ideal predictions. The results imply a tunable, geometry-dependent magnetic trapping mechanism that could inform low-cost rotor-based magnetic levitation devices and broaden practical magnetic-levitation applications.

Abstract

Recently, a novel magnetic levitation phenomenon involving two magnetically equivalent neodymium permanent magnets has been reported. In this work, we propose that this system functions as a scaled-up analog of the Levitron. The key distinction is that the ratio $m_/μ_f$ becomes a function of the lateral displacement $δ_R$, and magnetic trapping no longer depends on the rotational speed of the levitating body as in a conventional Levitron. Furthermore, we demonstrate that stable trapping occurs when a specific constraint on the $δ_R$ parameter is satisfied, ensuring that the potential energy reaches a minimum at the equilibrium point.

Figures (11)

• Figure 1: FIG.(1a) depicts the coordinates of the rotor (R) and floater (f) relative to $r_f = z_R - z_f$. The red arrow illustrates the orientation of the magnetic moment attributed to the rotor $\mu_R$, while the blue arrow represents the corresponding magnetic moment of the floater $\mu_f$. FIG. (1b), represent the scheme for characterizing the float’s energy as a function of the coordinates $(z_f, z_R)$, which define the ground state $U(0)_{grav}$ of the float.
• Figure 2: Representation of the configuration of the magnets coupled to the rotor (R) for situations in which the lateral displacement $\delta_R$ corresponds to (a) $\delta_R = 0$ and (b) $\delta_R > \delta^c_R$.
• Figure 3: Contour plot obtained from Eq.(\ref{['eq81a']}) for $\delta_R = 0.1mm$. The contextualization of the curves behavior are described in the text.
• Figure 4: Plot obtained for Eq.(\ref{['eq81a']}) assuming $R_f = 0$ in the cases where $\delta_R =1mm$, $1.5mm$ and $1.75mm$ for spherical, cylindrical and cube floats represented by blue dot-dash curves. The dimensions and mass of the cylindrical and cube floats correspond to width and height of $5mm$, $m_f = 0.73gr$, while for a spherical float the respective measurements are diameter $7mm$, mass $m_f = 1.53gr$. The magnetic momentum of cylindrical(cube) floats and rotor are $\mu_f = \mu_R = 0.063 A.m^2$, while that of the spherical is $\mu_f = 0.18 A.m^2$, the contextualization of the curves behavior are described in the text.
• Figure 5: Plot obtained for Eq.(\ref{['eq81a']}) assuming $(R_f)_{min}$ for spherical float considering the data in Table 1 for $\delta_R=1.5mm$. The contextualization of the curves behavior are described in the text.