Abstract:
The symmetry properties of a magnetic levitation arrangement are exploited to produce spin-stabilized magnetic levitation without aligning the rotational axis of the rotor with the direction of the force of gravity. The rotation of the rotor stabilizes perturbations directed parallel to the rotational axis.

Description:
This application claims the priority under 35 U.S.C. §119(e)(1) of co-pending provisional application Ser. No. 60/612,593, filed Sep. 22, 2004 and incorporated herein by reference. 

   This invention was developed under Contract DE-AC04-94AL8500 between Sandia Corporation and the U.S. Department of Energy. The U.S. Government has certain rights in this invention. 

   FIELD OF THE INVENTION 
   The invention relates generally to levitating objects and, more particularly, to levitating objects in magnetic fields. 
   BACKGROUND OF THE INVENTION 
   Earnshaw&#39;s theorem implies that it is impossible to achieve stable static magnetic levitation in a static magnetic field. However, the discovery of the Levitron™ has shown that it is in fact possible for a spinning top to be in stable equilibrium in a static magnetic field. This phenomenon, referred to herein as spin-stabilized magnetic levitation has been widely analyzed in the literature. The Levitron™ itself is described in U.S. Pat. No. 5,404,062, incorporated herein by reference. In general, in conventional spin-stabilized magnetic levitation devices such as the Levitron™ the rotational (spinning) motion of the rotor overcomes lateral instability of the rotor in the magnetic field. The conventional spin-stabilized devices are axisymmetric, and are limited to rotation about a vertical axis, that is, an axis aligned with the direction of the force of gravity. 
   It is desirable in view of the foregoing to provide for spin-stabilized magnetic levitation that does not require alignment of the rotational axis with the direction of the force of gravity. 
   Exemplary embodiments of the invention use the symmetry properties of a magnetic levitation arrangement to produce spin-stabilized magnetic levitation without aligning the rotational axis with the direction of the force of gravity. The rotation of the rotor stabilizes perturbations directed parallel to the rotational axis. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  diagrammatically illustrates a spin-stabilized magnetic levitation arrangement according to exemplary embodiments of the invention. 
       FIG. 2  diagrammatically illustrates a spin-stabilized magnetic levitation arrangement according to further exemplary embodiments of the invention. 
       FIGS. 3 and 4  mathematically illustrate symmetry properties of the arrangements shown in  FIGS. 1 and 2 . 
       FIGS. 5 and 6  mathematically illustrate equilibrium implications of the symmetry properties of  FIGS. 3-6 . 
       FIGS. 7-20  mathematically illustrate a linear stability analysis of the equilibrium conditions of  FIGS. 7-10  according to exemplary embodiments of the invention. 
       FIG. 21  graphically illustrates curves that define upper and lower limits on the spin rate of the rotors in  FIGS. 1 and 2  according to exemplary embodiments of the invention. 
       FIG. 22  illustrates conditions associated with the curves of  FIG. 21 . 
       FIG. 23  illustrates Taylor series expansions of the magnetic potentials about the dipole locations in arrangement such as shown in  FIG. 2 . 
       FIG. 24  illustrates operations that can be performed to compute dynamical constants for magnetic levitation arrangements according to exemplary embodiments of the invention. 
       FIGS. 25 and 26  mathematically illustrate exemplary operations that can be performed to determine a configuration of magnets that will provide a desired set of dynamical constants according to exemplary embodiments of the invention. 
   

   DETAILED DESCRIPTION 
   Exemplary embodiments of the present invention control magnetic symmetries so that a spinning rotor experiences equilibrium of magnetic forces and torques in all directions except the vertical force of gravity direction. The rotor spins about an axis extending in a direction other than the vertical direction, and the spin stabilizes the axial instability of the rotor.  FIGS. 1 and 2  diagrammatically illustrate exemplary embodiments of magnetic levitation arrangements according to the invention that can produce the aforementioned magnetic equilibrium conditions. 
   In the example of  FIG. 1 , the magnets of the base magnet system  11  produce a magnetic potential that is symmetric with respect to reflections about the planes x=0 and y=0. In some embodiments, all magnets of the base magnet system  11  are dipoles located in a plane z=constant, and they all point in the z direction. A first dipole located at (x0, y0, z0) has companion dipoles located at points (+/−x0, +/−y0, z0). All of the magnets have their dipoles pointing in the same direction. The rotor  12  is axisymmetric and carries a system of magnets that produces a magnetic potential that is symmetric with respect to reflection about its midplane. (In the examples shown in  FIGS. 1 and 2 , the midplane of the rotor is parallel to the x=0 plane.) The rotor  12  has its center of mass at spatial coordinates x=0, y=0, and z=0, and its axis of symmetry points in the x direction. In some embodiments, the rotor  12  includes two dipoles located on its axis of symmetry, positioned symmetrically about its midplane, and pointing in opposite directions along the axis of symmetry. 
   In the example of  FIG. 2 , the magnets of the base system  21  produce a magnetic potential that is symmetric with respect to reflection about the plane y=0 and is antisymmetric with respect to reflection about the plane x=0. A first dipole located at (x0, y0, z0) has companion dipoles located at points (+/−x0, +/−y0, z0). The dipole located at (x0, −y0, z0) points in the same direction as the first dipole, and the dipoles located at (−x0, +/−y0, z0) point in the opposite direction. In general, the base system  21  can have dipoles pointing in arbitrary directions, as long as appropriately reflected companion magnets are included. The rotor  22  is axisymmetric and carries a system of magnets that produces a magnetic potential that is antisymmetric with respect to reflection about its midplane. The rotor  22  is physically positioned in the magnetic field in the same manner described above with respect to the rotor  12  of  FIG. 1 . In some embodiments, the rotor  22  includes two dipoles located on its axis of symmetry, positioned symmetrically about its midplane, and pointing in the same direction along the axis of symmetry. 
   In each of the arrangements of  FIGS. 1 and 2 , due to the symmetries of the respective configurations, there are no forces in the x and y directions when the rotors are positioned as shown in the magnetic field. Similarly, there are no torques on the rotor. Equilibrium in the z direction can be obtained by suitably adjusting the weight of the rotor and/or the strengths of the magnets so that the force in the z direction balances the force of gravity G, which is assumed to act in the z direction. 
   Earnshaw&#39;s theorem implies that the equilibrium position must be unstable if the rotor is not spinning. Analysis of a spinning rotor in configurations such as shown in  FIGS. 1 and 2  reveals that the equations for perturbations in the lateral (e.g., y or z) directions decouple from the equations for angular perturbations and perturbations in the axial direction. This implies that it is not possible to spin-stabilize perturbations in the y and z directions, but that it is possible to spin-stabilize perturbations in the axial direction (parallel to the rotational axis of the rotor). This latter possibility contrasts with conventional systems where a rotor spins about a vertical axis in an axisymmetric field. In these vertical spin axis systems, is not possible to spin-stabilize perturbations in the axial direction. Rather, the vertical spin axis systems operate to spin-stabilize lateral perturbations. 
   In general, exemplary embodiments of the invention assume that: (1) the rotor and its magnets are axisymmetric; (2) in equilibrium, the rotor is aligned with its axis of symmetry in the x direction, and spins about the x-axis  13 ; and (3) the center of mass x of the rotor is spatially located at the point (0, 0, 0). As indicated above,  FIGS. 1 and 2  illustrate exemplary arrangements that meet these assumptions, and the following discussion is generally applicable to both  FIGS. 1 and 2 . 
   Because the rotor is axisymmetric, the energy of the rotor in an arbitrary magnetic field can be written as shown at  31  in  FIG. 3 , where  x  is the center of mass of the rotor, and  d  is a unit vector pointing in the direction of the axis of symmetry. The energy satisfies equation  32  of  FIG. 3 , which represents the Laplacian of the energy U with respect to the variable  x . The energy of a system where the potential is antisymmetric with respect to reflections about the x-axis satisfies the symmetry properties shown in  FIG. 4 . The energy of a system where the potential is symmetric with respect to reflections about the x-axis also satisfies the symmetry properties of  FIG. 4 . 
   Assuming a base system and rotor that satisfy the symmetry properties of FIG.  4 , equilibrium conditions can be identified. In particular, it can be shown that, if the rotor is placed so that its center of mass is at (0, 0, 0) and its axis of symmetry points in the direction  d =(1, 0, 0), then there is no torque on the rotor and the only component of force is in the z direction. Appropriate adjustment of the rotor weight and/or the strengths of the magnets can balance this magnetic force with the force of gravity. 
   It can be demonstrated that the force and torque on the rotor are given by the equations of  FIG. 5 . As shown, the force  51  depends on the gradient with respect to  x , and the torque  52  depends on the gradient with respect to  d . 
   The symmetry properties of the energy for both the antisymmetric (e.g.,  FIG. 2 ) and symmetric (e.g.,  FIG. 1 ) cases, are shown at  61  and  62  in  FIG. 6 . When the rotor is placed symmetrically in the field, the magnetic forces Fx and Fy in the x and y directions satisfy equations  63  and  64 . To show that the torques vanish, substitute x=y=0 into the symmetry property  41  of  FIG. 4  to get the result shown at  65  in  FIG. 6 . This shows that the energy at x=y=0 is an even function of dy and dx, so the derivatives with respect to dy and dx must vanish. Using the relation shown at  52  in  FIG. 5 , the expression  66  of  FIG. 6  can be seen. It can be seen from  63 - 66  that, if the rotor is positioned so that its center of mass is at x=y=0, and so that its axis of symmetry points in the x direction, there will be no forces in the x or y directions, and no torques at all. 
   The kinematics of the rotor can be described in a manner similar to that employed by Genta et al, in  Gyroscopic Stabilization of Passive Magnetic Levitation , Meccanica, 34 (1999), pp 411-424, incorporated herein by reference. In this description, the coordinates (x, y, z) refer to coordinates fixed in space, and the rotor is assumed to be axisymmetric, with moments of inertia I 3  about the axis of symmetry, and I 1  about the other two principal axes. Assume that the rotor is oriented by rotating about the z-axis by θ, the y-axis by φ, and then the x-axis by ψ. It can be shown that, if the rotor is spinning about the x-axis with angular velocity ω 0 , then a small perturbation to this state gives the approximate angular momenta shown in  FIG. 7 . The expression for Ly has two terms. The first term is the angular momentum that would occur if ω 0 =0 and the rotor were spinning about the y-axis. The second term is the angular momentum that would occur if the rotor kept spinning about the axis of symmetry with angular velocity ω 0 , but was slowly tilted by an amount θ about the x-axis. As a result of this tilting, some of the angular momentum that was initially in the x direction gets projected onto the y-axis. A similar interpretation can be given for Lz, the approximate angular momentum in the z direction. 
   The linearized equations of motion can be written as shown in  FIG. 8 . In this linear approximation, the forces and torques are linear functions of x, y, z, θ, and φ. The vector  d =(1, θ, −φ). It can be shown that, in the linear approximation, the forces and torques are derivable from a quadratic potential. Moreover, the symmetry properties show that many of the terms in the quadratic potential must be missing. It can therefore be shown that the linearized equations of motion are of the form shown in  FIG. 9 . Note that the equations for y and z decouple from the other equations. This means that A 1  and A 2  must both be greater than zero in order to have stability. In other words, the system would have to be stable with respect to lateral perturbations if the rotor were not spinning. Equation  32  of  FIG. 3  (or Earnshaw&#39;s theorem) implies that A 1 +A 2 =A, so the system must be unstable with respect to axial perturbations if the rotor is not spinning. 
     FIG. 10  introduces dimensionless variables x and t. Dimensionless equations  101 - 103  are written in terms of the dimensionless variables (after dropping the hats for convenience). Equations  101 - 103  also introduce the dimensionless parameters  104 - 107  of  FIG. 10 . 
   Turning now to an analysis of the stability of the linearized dynamical equations  101 - 103 , if solutions of the form  111  in  FIG. 11  are assumed, this leads to the characteristic polynomial  112 . This can be expanded to get equation  121  of  FIG. 12 , with q from  121  given at  122 . 
   In order for the system to be stable, all of the roots of equation  121  must be real and positive. Descarte&#39;s theorem implies that for an equation of the form z 3 +px 2 +qz+r=0 to have all real and positive roots, it is necessary that p&lt;0, q&gt;0, and r&lt;0. Furthermore, if all of the roots are real, then these conditions are both necessary and sufficient conditions for all of the roots to be positive. This, along with the aforementioned condition that A&gt;0 gives the conditions for stability shown in  FIG. 13 . The condition  131  is the requirement that A&gt;0 in order to have lateral stability. As with vertically oriented spin-stabilized magnetic levitation, there are upper and lower limit values of Ω for stability. 
   The parameters used in the equations of  FIG. 13  are defined at  141 - 144  of  FIG. 14 , and it is assumed that the parameters  141 - 143  are all large. Substituting  141 - 144  into equation  112 , multiplying by ε 4 , and setting ε=0, the equation of  FIG. 15  is obtained. This gives two roots of the sixth order polynomial  121  in  FIG. 12 . There are only positive solutions to a σ 2  if the conditions  161  and  162  of  FIG. 21  are met. The other four roots can be obtained by assuming the relationship at  163 . This results in equation  164  which, after factoring out the leftmost factor, yields equation  165 . Equation  165  is the characteristic equation for a spinning rotor in a harmonic potential. Application of the quadratic equation shows that, in order for equation  165  to have all real roots, the five conditions designated at  166  must be met. All of the conditions at  166  can be achieved by selecting a suitably large value of Ω. 
   These stability conditions can be understood by recognizing that, if the parameters  141 - 143  are large, and the system is not responding too quickly, equation  101  ( FIG. 10 ) implies the relationship shown at  171  in  FIG. 17 . This is equivalent to saying that, as the rotor moves around, it orients itself so that there is no torque on it. This gives the expression  172  which, when substituted into equation  101 , yields equation  173 . This will be a stable harmonic oscillator, subject to the condition  174 . This is the first asymptotic stability condition. In order to satisfy condition  174 , the condition  175  must be met, which implies that the rotor would want to flip over in the absence of spin. 
   The second asymptotic stability condition is that the rotor must spin fast enough to avoid flipping over. To analyze this mode, assume that σ is of order 1/ε. In this case, equation  101  implies that x is small compared to φ. This means that x can be ignored when solving equations  102  and  103 . This is equivalent to considering a rotor spinning in a potential where the translational energy is ignored. 
   The asymptotic analysis presented above does not predict the existence of an upper spin rate limit. In order to predict an upper limit, assume again that the parameters  141 - 143  of  FIG. 14  are large. It can be shown that, if Ω is too large, the eigenvalues that are of order one will eventually become unstable. Assuming that σ is order unity, the eigensystem can be approximated by the equations of  FIG. 18 , which are obtained by ignoring the second derivatives of θ and φ in  FIG. 10 .  FIG. 18  implies equation  191  of  FIG. 19 , a quadratic equation for σ 2 . In order for equation  191  to have positive real roots, the conditions shown at  192  must be satisfied. This gives a quadratic equation in z whose roots are defined at  193 . In order to have real roots, either z&lt;z− or z&gt;z+. However, if z&gt;z+, the other inequalities necessary for positive real roots cannot be satisfied. It therefore follows that the condition  194  must be satisfied. This is the asymptotic prediction for the upper spin rate limit. Assuming that that the parameters  141 - 143  are of order 1/ε 2 , then the upper limit on the spin rate is also of order 1/ε 2 . On the other hand, the lower spin rate limit is on the order of 1/ε. Accordingly, as ε becomes smaller, the ratio between the upper and lower spin rate limits can become very large. 
   Collecting now the results from the foregoing asymptotic stability analysis, and assuming the conditions shown at  201  in  FIG. 20 , the necessary and sufficient conditions for stability are shown at  202 . Again, with the parameters  141 - 143  of order 1/ε 2 , the upper spin rate limit is of order 1/ε 2 , and the lower spin rate limit is of order of 1/ε. This shows that the ratio between the upper and lower spin rates can be increased as desired, by increasing the values of the parameters  141 - 143  but holding the ratios therebetween fixed. 
     FIG. 21  graphically illustrates the above-described asymptotic estimates for the upper ( 211 ) and lower ( 212 ) spin rate limits. Also shown by broken line are more exact upper ( 213 ) and lower ( 214 ) spin rate limits, as computed using numerical methods. In particular, each curve of  FIG. 21  represents a limit on the spin rate Ω as a function of 1/ε, under the conditions specified in  FIG. 22 . 
   In some embodiments, the exemplary procedure described below is used to compute the dynamical constants A 1 , A 2 , B, C 1 , and C 2  that are needed for stable equilibrium with a given magnet configuration. For simplicity of exposition, it is assumed that the magnets on the rotor can be approximated by dipoles, although the analysis can be extended in a straightforward (though somewhat tedious) manner to magnets approximated as combinations of dipoles, quadrapoles, and octapoles. Assume, for example, that the rotor is positioned within a base magnet system that produces an antisymmetric potential (e.g., as in  FIG. 2 ). That is, the magnetic potential f(x, y, z) satisfies both f(x, y, z)=f(x, −y, z), and f(x, y, z)=−f(−x, y, z). Also assume that when the rotor is oriented in its equilibrium position, it has dipoles positioned at the points (+/−δ/2, 0, z0), both of magnitude M R , and both pointing in the direction (1, 0, 0). The dynamical components in this example are computed under the assumption of a single pair system of dipoles on the rotor. (With more than one dipole system, the overall dynamical constants can be computed as sums of the respectively corresponding dynamical constants computed for each dipole system.) 
   In order to compute the forces and torques that act upon the rotor as it gets displaced from its equilibrium position, the Taylor series expansions of the magnetic potentials about the points (+/−δ/2, 0, z0) are computed (e.g., up to the cubic terms). The Taylor series for both points are shown in  FIG. 23 . 
     FIG. 24  illustrates in detail at  2401 - 2411  exemplary operations for computing the dynamical constants and the lift according to the exemplary embodiments of the invention. The procedure defined by the illustrated operations can be readily implemented using commercially available software such as Mathematica. The operations illustrated at  2401 - 2411  are self explanatory. 
   Although the foregoing example of  FIGS. 23 and 24  relates to an antisymmetric configuration such as shown in  FIG. 2 , the same results can also be obtained for a symmetric configuration such as shown in  FIG. 1 . With the symmetric configuration, the Taylor series expansion about the point (δ/2, 0, z0) is the same as for the antisymmetric configuration, and the expansion about the point (−δ/2, 0, z0) is exactly opposite of that obtained for the antisymmetric configuration. If the fields are defined using the Taylor series expansion about (δ/2, 0, z0), the dynamical constants have the exact same values as those given for the antisymmetric configuration. 
   In some embodiments, the exemplary procedure described below is used to determine a particular configuration of magnets that will produce a desired set of the dynamical constants. The example described below assumes a system having a magnetic potential that exhibits reflectional symmetry about the x-axis. Assume, for example, an overall base system that consists of 4N dipoles (N systems of four dipoles each) which all point in the z direction. The positions of the dipoles are given by  251  in  FIG. 25 , and the magnetizations of the dipoles are given by  252 . Each value of the index i corresponds to a four magnet system symmetrically positioned in the overall base system. For each value of i, the dynamical parameters A 1 ( i ), A 2 ( i ), A(i), B(i), C 1 ( i ), C 2 ( i ), and L(i) can be computed for di=1. The values of the dynamical parameters for the whole system can be obtained by summing over the different sets of magnets multiplied by the strengths of the dipoles. An example of this is shown in  FIG. 26 . If there are six or more systems of magnets, then there are six equations in six unknowns (recall that A is known in terms of A 1  and A 2 ), the strengths di can be chosen as necessary to produce any desired values of the dynamical parameters. 
   Referring again to  FIGS. 1 and 2 , these illustrate specific examples of more general symmetry configurations according to the invention. Two general symmetry configurations according to the invention are now described. Both general symmetry configurations use an axisymmetric rotor, with M systems of magnetic dipoles on the axis of symmetry of the rotor. The mass distribution of the rotor is assumed to have reflection symmetry about its midplane. All of the rotor&#39;s dipoles point in the direction of the rotor&#39;s axis of symmetry. For purposes of this description of the general configurations, the equilibrium position of the rotor is defined so that the center of mass is at (0, 0, 0) and the axis of symmetry is oriented in the x direction (see also  FIGS. 1 and 2 ). When the rotor occupies this position, the kth system of dipoles on the rotor has dipoles at (+/−δk, 0, 0). Both general symmetry configurations use a base that contains N systems of dipoles. The kth system in the base contains four dipoles, at  p 1k=(ak, bk, ck),  p 2k=(−ak, bk, ck),  p 3k=(ak, −bk, ck), and  p 4k=(−ak, −bk, ck). In this configuration, if the dipole at  p 1k of the kth system in the base has a magnetic dipole moment of (pk, qk, rk), then the remaining dipoles at  p 2k,  p 3k, and  p 4k of the kth system have respective magnetic dipole moments of (−pk, qk, rk), (pk, −qk, rk), and (−pk, −qk, rk). (Note that N=1 in  FIGS. 1 and 2 .) 
   For a symmetric rotor configuration, when the rotor is in its equilibrium position, its kth dipole system has dipoles at (+/−δk, 0, 0) with dipoles moments of (+/−mk, 0, 0). That is, the symmetrically placed dipoles are pointing in opposite directions. In this configuration, if the dipole at  p 1k of the kth system in the base has a dipole moment of (pk, qk, rk), then the remaining dipoles at  p 2k,  p 3k, and  p 4k of the kth system in the base have respective dipole moments (−pk, qk, rk), (pk, −qk, rk), and (−pk, −qk, rk). (Note that pk=qk=0 in  FIG. 1 .) 
   For an antisymmetric rotor configuration, when the rotor is in its equilibrium position, its kth dipole system has dipoles at (+/−δk, 0, 0) with dipoles moments of (mk, 0, 0). That is, the symmetrically placed dipoles are pointing in the same direction. In this configuration, if the dipole at  p 1k of the kth system in the base has a dipole moment of (pk, qk, rk), then the remaining dipoles at  p 2k,  p 3k, and  p 4k of the kth system in the base have respective dipole moments (pk, −qk, −rk), (pk, −qk, rk), and (pk, qk, −rk). (Note that pk=qk=0 in  FIG. 2 .) 
   Having now described symmetry configurations according to the invention in general terms, a detailed example of a specific design configuration according to the invention is set forth in the Appendix. 
   Exemplary features of the invention described in detail above are summarized hereinbelow. 
   1. If the system is constructed according to the above-described symmetry configurations, then equilibrium of forces and torques is achieved except for equilibrium in the z direction. 
   2. For a given equilibrium condition, the dynamics governing small displacements from the equilibrium are determined by the following parameters: the mass of the rotor; the moment of inertia I 3  of the rotor about the axis of symmetry; the moment of inertia I 1  of the rotor about the axes perpendicular to the axis of symmetry; the rotor spin rate ω; and the dynamical constants A 1 , A 2 , C 1 , C 2 , A, and B. 
   3. For a given configuration of magnets, the dynamical parameters can be determined either theoretically, or by numerically computing the derivatives of the forces and the torques as the center of mass and orientation of the rotor are changed. In particular, A 1  is the derivative of the force in the y direction with respect to a change in the y position of the rotor, A 2  is the derivative of the force in the z direction with respect to a change in the z position of the rotor, and A is equal to A 1 +A 2 . C 1  gives the derivative of the torque about the z-axis with respect to the angle θ of the rotation of the rotor about the z-axis. C 2  gives the derivative of the torque about the y-axis with respect to the angle φ of the rotation of the rotor about the y-axis. B gives the derivative of the force in the x direction with respect to the angle φ of the rotation of the rotor about the y-axis. 
   4. Given the dynamical parameters described above, the stability or instability of the system can be determined therefrom. For some configurations of magnets, the system is unstable for all values of spin rates. For some configurations, there is a range of spin rates within which stability is achieved. 
   5. The conditions for stability are easily expressed in terms of the aforementioned dimensionless parameters Γ 1 , Γ 2 , Λ, and Ω. 
   Although exemplary embodiments of the invention have been described above in detail, this does not limit the scope of the invention, which can be practiced in a variety of embodiments.