This video gives a stroboscopic view of a magnetic bound state dynamics. Actually it is a standard video recording while the shuttle speed set high due to intense illumination of the scene. Stroboscopic view is obtained by adjusting speed of the motor close to 60 rev/sec.
In the below video from a similar setup the syncronized angular motion of the floating magnet with the driving magnet can be clearly seen. In this motion the top pole (S) of the floating magnet leans toward the S pole of the driving magnet all the time despite it get repelled.
This figure below shows an instance from the above video where poles of magnets are marked with letters in white. We can see that the top magnet is not aligned by 90 degrees to the shaft but tilted about 10 degrees. This alignment can be thought as superposition of two virtual magnets, one orthogonal to the shaft (radial) and the other (axial, blue) aligned with the shaft. The virtual radial magnet has strength of the real magnet by the cosine of the tilt angle and the axial one by the sine.
Time average of the radial magnet vanishes but the axial (blue) one resides. So there is a good reason for another magnet nearby to get aligned to the blue virtual magnet and get attracted. This is the case of the floating magnet beneath, however it does not stay fully aligned since it oscillates.
The cause of this oscillation is the rotation of the top magnet, precisely the virtual radial magnet. This magnet exerts continually spatially varying torque to the floating magnet causing a driven harmonic motion. This rotates the magnet orientation and can be called conical motion. In this picture instance the torque vector is perpendicular to the page and trying to rotate the floating magnet clockwise.
This motion can be fully synchronized with the top magnet under suitable conditions. This is the case in this video where the orientation of these magnets are locked to each other. That is looking through a camera which co-rotates with the driving magnet, all magnetic orientations, fields, torques, forces are standing still. Only the floating magnet can spin around its dipole axis freely. What is quite interesting here is the polar alignments of magnets.
We are seeing that the S pole of the bottom magnet leans toward the S pole of the top magnet despite it getting repelled. This appears counterintuitive however it is the result of the driven harmonic motion (DHM) and called phase lag.
In DHM, the displacement and the driving force can be in the same or in the opposite direction depending on the frequency of the driving force. Every harmonic motion can be associated with a natural frequency. This frequency is determined by the inertial factor of the object under harmonic motion and the spring constant of the force applied to this object. Inertial factor is the moment of inertia and we use spring constant related to torque and angular displacement here.
Using the spring constant term C and a mass m, the natural frequency ω₀ can be calculated by the relation ω₀² = C/m. Although the spring constant term can only be applied here by approximation, we only need to estimate the natural frequency here as an order of magnitude.
Phase lag is the phase of the motion with respect to the driving force or torque. If we neglect frictions or other causes of damping in a DHM, phase lag is zero when driving frequency is below than natural frequency and becomes 180° when it is above. Here the motor speed is likely well above the natural frequency and the phase lag is 180°. For this reason the S pole of the floating magnet is looking at the S pole of the top magnet all the time.
The net effect is the phrase lag which keeps the floating magnet orientation stable with respect to the rotating magnet favoring repulsion. Such an orientation is impossible to obtain between static magnets when one magnet is free to move.
In summary the tilt of the rotating magnet causes an attraction to the floating magnet which is balanced by the weight of the magnet and by the magnetic repulsion explained here. We can call this equilibrium as a magnetic bound state.
This equilibrium is stable because the repulsion factor increases two times faster than the attraction when magnets get closer. This is because the amplitude of angular motion increases when magnets get closer resulting in a larger conical motion.
Actually the strength of the repulsion is approximately proportional to the amplitude of this oscillation. For this reason, increasing the motor speed (the frequency) reduces oscillation amplitude, causing the repulsion factor to reduce. This way floating magnet can approach the driving magnet more and can find a new equilibrium where attractive and repulsive interactions become stronger than before. This reduced air gap allows the interaction to carry more loads.
In detail, the dependence of the oscillation amplitude on the frequency is approximately inverse square when this frequency is significantly larger than the natural frequency of the system. The related equation can also be found in common texts on driven harmonic motion. Only we deal here with angular oscillation, torque and moment of inertia instead of translational motion, force and mass. It should be noted natural frequency term ω₀ is mostly used in textbooks in related equations. This property can be easily identified and observed in textbook examples however not in these magnetic trapping solutions.
This image shows the magnetic field of a similar setup. The magnetic force vector is horizontal and it rotates around the vertical axis (motor shaft direction) This means its time average is zero. This rotating force causes the FM to perform small circles with a sub mm radius. This circular motion has also characteristics of the DHM. That is the displacement of the body can be in the opposite direction of the force it receives and bring repelling poles of magnets even closer.
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We also see that large floating magnets or magnets fixed to rigid loads oscillate less. This is because oscillation amplitude is also dependent inversely to the moment of inertia of the floating object with respect to its pivoting center. This pivoting center can be different from the dipole center and the center of mass when floating bodies consist of a magnet attached to an inert mass.
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