== Model and Specification ==

{{attachment:fig_sus_core_new.JPG}}

The magnets have typical remanence of 1T and recoil permeability of 1.05. 

== Vertical Direction ==

The following figure shows the resulting magnetic force in z-direction as a function of the gap
between two magnets. The dots are numerical results, and they are fitted by simple 
polynomial functions as shown by the red solid curve. This recovers the previous 
qualitative results based upon the current-loop arguments.  

{{attachment:F_z_1x1_mag.JPG}}

As we can see, the DC magnetic force is around 20 N near the maximal-force point. 
It allows us to levitate a ~2 kg test mass. 

The corresponding rigidity around the maximal-force point is shown by the red curve in the figure below. It 
is obtained by taking the derivative of the fitting polynomial function for the DC magnetic force. 
As expected, the rigidity can be arbitrarily low if the suspended magnet can be located close to
the maximal-force point.  

{{attachment:K_z_1x1_mag.JPEG}}


== Horizontal Direction ==

The results for the y-direction are shown by the following two figures:

{{attachment:F_y_1x1_mag.JPG}}

{{attachment:K_y_1x1_mag.JPG}}

In producing the above results for y-direction, we have chosen the optimal gap between the two magnets.  

The Comsol file and m-files for producing the above results are

[[attachment:model_1x1_mag_force.mph]]

[[attachment:model_1x1_mag_force_y.m]]

[[attachment:model_1x1_mag_force_z.m]]

== Torque ==

The corresponding torque of the suspended magnet and its rigidity around y-axis are shown by the figures below: 

{{attachment:torque_1x1_mag.JPG}}

{{attachment:K_theta_1x1_mag.JPG}}

As we can see, the torque near the maximal-force point is almost zero. In addition, since the 
two magnetic dipoles tend to align each other, it is also stable. Due to the symmetry, the 
situation with torque in the x-direction is the same. Therefore, this simple case already 
shows the possibility to achieve low resonant frequencies in all six degrees of freedom of the suspended
magnet. This is one of the most attractive features of this scheme. 

The Comsol file and m-file for evaluating torque is the following:

[[attachment:model_1x1_mag_torque.mph]]

[[attachment:model_1x1_mag_torque.m]]