From: crb7q@kelvin.seas.Virginia.EDU (Cameron Randale Bass)
Subject: Re: Passive Magnetic Levitation (was: Science Fair Project)
Organization: University of Virginia
Date: Fri, 11 Dec 1992 20:28:53 GMT

In article <1992Dec11.170337.22897@asl.dl.nec.com> terry@asl.dl.nec.com writes:
>In article <BCR.92Dec11013519@hfl3sn02.cern.ch>
>bcr@cernapo.cern.ch (Bill Riemers) writes:
>
>> In article <1992Dec10.221950.13548@asl.dl.nec.com>
>> terry@asl.dl.nec.com writes:
>>
>> | Wow, strong magnets!  But I cannot see how that could possibly be right --
>> | the [Bollinger] 3-pronger will either slide off in some fashion, or it
>> | won't.  There isn't enough stored energy in the system to make it flip,
>> | nor do the intense regions of the repulsive fields extend far enough out
>> | to perform such a feat.
>>
>> Not true.  If there is enough force to levitate it then there must also be
>> enough force to flip it.
>
>Sure -- in the absence of gravity, my 3-pronger will eventually flip over.
>
>However, when I said that there "isn't enough stored energy in the system to
>make it flip," I was referring to the energy needed to raise it out of the
>metastable state, not the net energy gain:
>
>            Metastable   Energy Needed to     Stable
>            Levitation    Rotate 3-Prong       Final
>           Energy State  Around horiz axis     State
>                                                       
>                                --                     
>                              -    -                   
>                             -      -                  
>                            -        -                 
>                           -          -                
>                          -             -              
>                        -                -            
>               -------                     -           
>                                             -         
>                                               ------  
>                                 
>     Peak_energy(theta) = f(arm_length) = ~ constant x arm_length
>
>Would you agree that this diagram and the ballpark equation under it describe
>the energetics of flipping my three-pronged device in gravity?  Note that if
>you assume the (levitated) magnets to have constant mass and that materials
>exist whereby stretching the length of the arms within plausible limits (say
>up to a meter) adds no significant mass to the starting mass of the system,
>you can *always* increase the flip-over stability of the system until the
>likelihood of a flip-over around a horizontal axis becomes vanishingly small.

      Yes, but then they still tip and torque and find the minimum in the
      field strenght and flop over.

>However, rotating the 3-prong device around a vertical axis (and/or moving
>moving it in the horizontal plane) also produces a curve like the one above,
>except that the "final" state just becomes removing the 3-prong and setting
>it down on the table.  This is the potential energy curve that I would argue
>is the relevant one for a 3-prong, because unlike the first curve there is
>no obvious, easy way to increase the height of the potential energy peak
>when going from the levitation state to the "flat on the table" state.

      There are six degrees of freedom.  You've pinned two.  There are now
      three translations and one rotation unaccounted for.  You can
      always arrange it so that you pin five degrees of freedom, and the last
      one will still cause it to flop.  The 'top' of your levitating magnet
      in the 'pizza box' is strongly attracted to the outer levitator
      magnets on the 'arms'.  My guess is that torsional perturbations
      do you in. The levitated object simply flops out of your carefully
      arranged situation and proceeds to low energy resting on the ground.

>But what if the arm between A and B is very, very long?  If B rises by the
>same amount, the resulting angular motion of A will be greatly reduced,
>and thus the torque instability (which for small angles is proportional
>to the angle to which the magnet's axis has been twisted away from vertical)
>will be greatly reduced, too.  Feed that back through the long leverage arm
>between A and B and the return lift to B will be also be rather drastically
>reduced.  You can get a general feel for what's going considering case of
>what would happen if infinitely rigid, infinitely light arms were extended
>to infinity; *any* twisting of *any* of the magnets would require infinite
>energy to accomplish, no matter how small the angle involved.

     I suppose if the arms were very very very long, then angular momentum
     considerations could arbitrarily slow the instability.  But that's
     not really fair is it?  An arbitrarily long rigid massless rod
     connecting two masses is going to require arbitrarily large amounts of 
     energy to rotate it, but that is not really connected with the stability
     of systems of permanent magnets.

>But in all real cases there is still *some* of this twisting-and-feedback
>effect, right?  Yes, but what I've had no luck at all if figuring out how
>such greatly attenuated responses could destabilize the system.  The
>problem is that once their impact falls to a level below the height of the
>levitation-to-table energy barrier (the one provided ring magnets), they
>no longe appear to be capable of "joining together" to make the overall
>setup unstable.  If that is true, a real, bonafide passive "pizza box"
>levitation arrangement should be attainable with the 3-prong.

     It's simply stability theory.  A system can be unstable to arbitrarily
     small perturbations, no matter what the time constant of the system
     response.  You are mixing the two concepts, the stability and the
     time constant of an unstable response.  

>Anyone notice any subtle instabilities, or errors in the above analysis?

     Only that six DOF systems containing only permanent magnets are not
     possible, no matter how clever the arrangement of magnets.  I'd love
     to be able to quote chapter and verse, but it appears I have
     no good references within arms reach on the subject (and I'm not 
     willing to wing it).  Maybe someone else has the time?

>(Dr. Carr, any comments?)
>
>Any experiments yet?  It might make a nice paper if it actually works...

     It's a lifetime project.  I wouldn't hold my breath on a paper.

     Have a nice day.

                           dale bass
-- 
C. R. Bass                                          crb7q@virginia.edu        
Department of Mechanical, 
     Aerospace and Nuclear Engineering
University of Virginia                              (804) 924-7926