稳定的磁悬浮陀螺Spin stabilized magnetic levitation

Introduction

The Levitron®1 ~Ref. 1! is a remarkable toy which levitates in air a 22-g spinning permanent magnet in the form of a small handspun top. The top is spun on a lifter plate on a permanent magnet base and then raised to the levitation height. The top floats about 3.2 cm above the base for over 2 min until its spin rate declines due to air resistance to about 1000 rpm. Unlike an earlier magnet toy which requires a thrust bearing plate to stabilize motion along one direction,2 the magnetic top floats freely above the base magnet and is fully trapped in three dimensions ~see Fig. 1!.

Levitron是一种了不起的玩具,它是在空中悬浮着的22 g旋转的永磁体,呈小手纺陀螺的形式。顶部在永久磁铁基座上的提升板上旋转,然后升至悬浮高度。顶部在底部上方约3.2 cm处浮动约2分钟,其旋转速度由于约1000 rpm的空气阻力而下降。而早期的磁铁玩具需要一个止推轴承板来稳定沿一个方向的运动,2,但这款磁性陀螺可自由漂浮在基础磁铁上方,并完全被困在三个维度中(见图1!)。

Since Earnshaw’s theorem of 18423 rules out stable magnetic levitation for static magnetic dipoles, it was not obvious to us how the Levitron worked. A simple theory of gyroscopic stability against flipping proposed by the manufacturer and others4,5 is not sufficient to explain the stability. Magnetic levitation of spinning permanent magnet tops was discovered by inventor Roy Harrigan who patented it in 286 Am. J. Phys. 65 ~4!, April 1997 © 1997 American Association of Physics Teachers 2861983.6 Harrigan persisted in his efforts even after being told by several physicists that permanent magnet levitation was impossible and that he was wasting his time.7

由于Earnshaw的18423定理排除了静态磁偶极子的稳定磁悬浮,因此我们对Levitron的工作原理并不了解。制造商和其他人[4,5]提出的一种简单的陀螺抗倒转稳定性理论不足以解释其稳定性。发明人Roy Harrigan在286 Am申请了专利,从而使旋转的永磁体顶部磁悬浮。 J.物理65〜4!,1997年4月©1997美国物理教师协会2861983.6即使几位物理学家告诉哈里根永久磁悬浮是不可能的,并且浪费了他的时间,哈里根仍然坚持自己的努力。7

Besides discovering spin stabilization Harrigan designed a square dishshaped base that established a suitable magnetic field configuration, made a top with the right rotational inertia, mass, and magnetic moment, found the small capture volume, and invented a means of moving the spinning top to the right location. The parameter space for successful levitation is quite small. Not much happened with the invention until 1993 when Bill Hones of Fascinations learned of Harrigan’s patent and saw a working prototype of the levitating top. Hones and Harrigan had a brief collaboration to make and market a levitating top toy but it soon ended.7,8

除了发现自旋稳定以外,哈里根还设计了一个正方形。碟形的基座建立了合适的磁场结构,使陀螺具有合适的旋转惯性,质量和磁矩,捕获量很小,并发明了一种将陀螺移动到正确位置的方法。成功悬浮的参数空间很小。直到1993年,Fastinations的Bill Hones学会了Harrigan的专利并看到了这种悬浮式陀螺的工作原型之后,这项发明才发生了很多事。 Hones和Harrigan进行了短暂的合作,制造并销售了一种悬浮式陀螺玩具,但很快就结束了。7,8

In 1994 Bill Hones and his father applied for a patent on a levitating top that used a square permanent magnet base, which was issued in 1995.4 The Levitron, made by Fascinations, has a square base magnet with a region of weaker or null magnetization in the center. The Hones’ patent states that levitation over a circular base magnet is not possible. We routinely use circular ring magnets which work at least as well as a square base.

1994年,比尔·霍恩斯(Bill Hones)和他的父亲在悬浮式陀螺上申请了一项专利,该悬浮式陀螺使用的是方形永磁体底座,该专利于1995年发布。4Fascinations生产的Levitron具有方形基础磁体,其磁化强度较弱或为零。 中央。 霍恩斯(Hones)的专利指出,无法在圆形底座上磁悬浮。 我们通常使用需要方形底座的圆形环形磁铁。

Our investigation included measurements of the commercial toy as well as modified experimental versions. We used air jets and then electromagnetic drives to counter the effects of air resistance and to spin the top faster. We also numerically integrated the equations of motion to determine the stability limits and compare to our calculations and experiments.

我们的调查包括对商用玩具的测量以及修改后的实验版本。 我们先使用空气喷射器,然后使用电磁驱动器来抵消空气阻力的影响,并使顶盖更快旋转。 我们还对运动方程进行了数值积分,以确定稳定性极限,并与我们的计算和实验进行了比较。

Our most interesting finding is that there is a maximum spin limit beyond which the top is unstable and cannot be confined. Understanding this feature is essential to understanding the actual trapping mechanism. While writing this paper, we became aware of a paper by Dr. Michael Berry, now published in the Proceedings of the Royal Society of London.9 He was kind enough to send us a preprint of his paper which we highly recommend.

我们最有趣的发现是,有一个最大的旋转限制,超过该旋转限制,顶部不稳定且无法限制。 了解此功能对于了解实际的稳定机制至关重要。 在撰写本文时,我们知道了迈克尔·贝里(Michael Berry)博士的一篇论文,该论文现已发表在《伦敦皇家学会学报》上。9他很友善地向我们发送了他的论文的预印本,我们强烈建议他这样做。

Our conclusions about the trapping mechanism are essentially the same as his. Berry develops the theory of the adiabatic invariant further than we do here. We would also like to thank Dr. Berry for reviewing an earlier draft of this paper and making helpful suggestions.

我们对稳定机制的结论与他的结论基本相同。 贝瑞比我们在这里进一步发展了绝热不变性的理论。 我们还要感谢Berry博士审阅了本文的早期草案并提出了有益的建议。

How it works

First, it is instructive to see how this trap for magnetic dipoles doesn’t work. It is not enough to simply stabilize the top/dipole against flipping. We can consider this the infinite spin case, whether the stability against flipping is provided by spin or by some mechanical arrangement. Assume the top’s magnetic dipole moment m is always oriented in the vertically downward 2z direction and the repulsive magnetic field B from the base magnet is primarily in the vertically upward 1z direction in the levitation region.

首先,了解此磁偶极阱如何工作很有启发性。 仅仅稳定顶部/偶极子以防翻转是不够的。 我们需要考虑无限旋转下去的情况,因此要靠旋转自身或某种机制来保持稳定。 假设顶部的磁偶极矩μ总是垂直向下指向-z轴,而基底磁铁的排斥磁场B总是垂直向上指向 +z轴。

The potential energy U is U52m–B1mgz5mBz1mgz. There are two conditions for stable levitation. The lifting force 2m(]Bz /]z) must balance the weight of the top mg and the potential energy at the levitation point must be a minimum. If the energy is a minimum it must have positive curvature in every direction or m(]2Bz /]xi 2).0, where the xi are x, y, and z. However, „2Bz50 at any point in free space so the energy minimum condition cannot be satisfied in all directions. Instead of a minimum there is a saddle point. This is just a consequence of the fact that the magnetic field in the trapping region is divergence and curl free.

于是势能U 就是U = – μ * B + mgz = μ*B_z + mgz。这要求

抬起的力 -μ(∂B_z/∂z) 必须与上面 mg的力相等,悬浮点的势能要最小。而如果要势能最小,则每个方向的curvature 要大于0,即

这里的 x_i 即x,y,z。然而,自由空间的任意点处的▽^2 B_z = 0,所以最小势能的情况得不到满足。但除了势能最小的情况,还有一个鞍点,这只是磁场在捕获区域无卷曲和散度导致的结果。

For completeness we note a second way that the trap does not work. We considered that the trap might work by strong focusing. If the top and/or base had nonuniform magnetization, the spinning might create the appropriate timedependent force to be a stable solution of the Mathieu equation. Measurements of the nonuniformities, the top’s inclination, and rotation showed that any focusing forces were too small by many orders of magnitude. Replacing the commercial square-magnetized base with a cylindrically symmetric ring magnet does not degrade the confinement at all, contrary to what one would expect if strong focusing was the trapping mechanism. The gyroscopic action must do more than prevent the top from flipping.

为了完整起见,我们注意到陷阱不起作用的第二种方式。 我们认为陷阱可以通过strong focusing 作用。 如果顶部和/或底部的磁化不均匀,则旋转可能会产生适当的时间依赖性力,从而成为Mathieu方程的稳定解。 对不均匀度,顶部倾斜度和旋转度的测量表明,任何 focusing forces 都小了几个数量级。 用圆柱对称的环形磁体代替商品化的方形磁化基座根本不会降低限制,这与如果 strong focusing 是捕获机制所期望的相反。 陀螺的作用必须要不仅限于防止陀螺翻转。

It must act to continuously align the top’s precession axis to the local magnetic field direction ~see Fig. 2!. Under suitable conditions, the component of the magnetic moment along the local magnetic field direction is an adiabatic invariant. When these conditions are met, the potential energy depends only on the magnitude of the magnetic field and gravity. While each component of the magnetic field must satisfy Laplace’s equation ~i.e., „2Bz50!, the magnitude of the magnetic field does not. This allows the curvature of the potential energy to be concave up ~and not a saddle point! at the levitation height. Properly understood, the trap mechanism is similar to magnetic gradient traps for neutral particles with a quantum magnetic dipole moment.

它必须起到使顶部进动轴与本地磁场方向连续对齐的作用,见图2! 在合适的条件下,沿局部磁场方向的磁矩分量是绝热不变的。 当满足这些条件时,势能仅取决于磁场和重力的大小。 尽管磁场的每个分量都必须满足拉普拉斯方程〜即 ▽^2 B_z = 0 !,但磁场的大小却不满足。 这样可以使势能的曲率向上凹入而不是一个鞍点! 在悬浮高度。 也就是说,陷阱机制类似于具有量子磁偶极矩的中性粒子的磁梯度陷阱。

Such traps were first proposed and used for trapping cold neutrons10 and are currently used to trap atoms,11 including recent demonstrations of Bose– Einstein condensation. The spin magnetic moment of a particle such as a neutron along the magnetic field direction is Fig. 1. General configuration for spin stabilized magnetic levitation. The commercial Levitron actually has a solid square base uniformly magnetized except for a circular region in the center. Ring magnets work fine despite some patent claims that it is impossible to levitate over circular magnets ~Ref. 4!. Fig. 2. As the top moves off center, its precession axis orients to the local field direction. Without this reorientation, radial confinement would be impossible at the levitation height. 287 Am.

这类陷阱是最早用于俘获冷中子,目前被用来俘获原子,包括最近玻色-爱因斯坦凝聚的证明。 沿磁场方向的中子等粒子的自旋磁矩为图。自旋稳定磁悬浮的一般结构。 商业Levitron实际上具有一个均匀的方形底座,除了中心的圆形区域外,该底座均被磁化。 尽管有些专利声称环形磁铁不能悬浮在圆形磁铁上,但环形磁铁仍然可以正常工作。 图2.当顶部偏离中心时,其进动轴将朝向局部磁场方向。 如果没有这种重新定向,则在悬浮高度处将无法进行径向限制。

J. Phys., Vol. 65, No. 4, April 1997 Simon, Heflinger, and Ridgway 287an adiabatic invariant. If the field does not change too rapidly or go through zero allowing a spin flip, the spin magnetic moment along the magnetic field direction is constant. The potential energy then depends only on the magnitude of the magnetic field. Since localized magnetic field minima are allowed ~isolated maxima are prohibited! by the laws of magnetostatics, a trap for antialigned dipoles is possible. Spin polarized particles or atoms seek the weak-field position in magnetic gradient traps. Another example of a similar adiabatic invariant is the magnetic moment of a charged particle spiraling along a magnetic field line. Here again, if the field changes slowly, the magnetic moment due to the particle orbit perpendicular to the field is constant.

如果磁场变化不是太快,或没有经过零导致翻转,则沿磁场方向的自旋磁矩将保持恒定。 然后,势能仅取决于磁场的大小。 由于允许局部磁场最小值,所以禁止隔离最大值! 根据静磁定律,可能会形成反取向偶极子阱。 自旋极化的粒子或原子在磁性梯度陷阱中寻找弱场位置。 相似的绝热不变量的另一个例子是带电粒子沿磁场线螺旋形的磁矩。 同样,如果磁场变化缓慢,则由于垂直于磁场的质点轨道而产生的磁矩是恒定的。

A charged particle can be trapped in the low field part of a magnetic mirror. We make two simplifying assumptions. First we assume that the top is a magnetic dipole whose center is also the center of mass. The position of the center of mass and the dipole are located at the same coordinates r. Second, we assume the ‘‘fast’’ top condition that the angular momentum is along the spin axis of the top which also coincides with the magnetic moment axis.

带电粒子可能被捕获在磁镜的低场部分。 我们做出两个简化的假设。 首先,我们假设顶部是一个磁偶极子,其中心也是质心。 质心和偶极子的位置位于相同的坐标r上。 其次,我们假设“快”顶条件是角动量沿顶的自旋轴,该顶角也与磁矩轴重合。

~We relax the fast top condition in the computer simulation code described in Appendix B.! That is, the angular momentum L5Iv(m/m). Here, I is the rotational inertia of the top around the spin axis, v is the constant angular spin frequency, and m is the magnetic moment. m5umu and is constant. The spin v can have a plus or a minus sign due to the two possible spin directions, parallel or antiparallel to m, respectively. The sense of the angular momentum does not affect the stability of the top, only the sense of the precession. The torque and force equations that describe the motion of the top ~ignoring air resistance and other losses! are

〜我们放松了附录B中描述的计算机仿真代码中的快速最高条件。 即:

角动量 L = Iw ( μ / μ )

此处,I是顶部围绕旋转轴的旋转惯量,w是恒定的旋转角频率, μ 是磁矩。 | μ / μ | 是常数。 由于两个可能的旋转方向相反,即平行或反平行于 μ ,则自旋w可以具有正号或负号。 角动量不会影响顶部的稳定性,只会影响进动感。 描述顶部运动的扭矩和力方程式(忽略了空气阻力和其他损失):

磁场是位置B(r)的函数,而磁矩取决于位置B(r) 和 μr,t)。

这两个方程中上方的方程顶部旋转轴与磁场B的 角进动频率 方向有关。

It is important to note that the precession frequency is inversely proportional to the spin frequency. Although we have assumed that the top is ‘‘fast,’’ if it is too fast, the precession frequency will be too slow to keep the top oriented to the local magnetic field direction. This is the origin of the upper spin limit.

重要的是要注意进动频率与自旋频率成反比。 尽管我们假设顶部是“快速”,但如果过快,进动频率将太慢而无法使顶部保持在局部磁场方向上。 这是自旋上限。

Equation ~1! also says that to lowest order, the component of the magnetic moment along the local magnetic field direction is a constant which we can call mi . We consider the case shown in Figs. 1 and 2 where mi is antiparallel ~repulsive orientation! to B. The potential energy of the top is

等式1! 还说到最低阶,沿着局部磁场方向的磁矩分量是一个常数,我们可以称 μ ||
。 我们考虑图1和2所示的情况, μ || 是反平行于B的。顶部的势能是

We can expand the magnetic field around the levitation point as a power series for our cylindrically symmetric geometry

我们可以将悬浮点周围的磁场扩展为圆柱对称几何的幂级数:

At the levitation point, the expression in the first curly braces must go to zero. The magnetic field gradient balances the force of gravity

在悬浮点,第一个大括号中的表达式必须为零。 磁场梯度平衡了重力S = -(mg) / μ 。

if the ratio of the mass to the magnetic moment m/m is correct. This ratio is adjusted by adding small weights to the top. For the potential energy to be a minimum at the trapping point, both K and $@(S/2) 2/B 0K#21% must be positive. The energy well is then quadratic in both r and z and approximates a harmonic oscillator potential. Thus, the trapping condition at the levitation point is

如果质量与磁矩之比m / μ 是合适的话,那么通过在顶部增加较小的重量就能调整比率。 为了使势能在陷阱点处最小,K和{[(s / 2)^2 / B_0 K] – 1}都必须为正。 然后,能量阱在r和z上均为二次方,并且近似于谐波振荡器的电势。 因此,在悬浮点的俘获条件为

If the magnetic moment was not free to orient to the local field direction as it moved off center ~see Fig. 2!, the term in the second curly braces would be only $21%, and the top would be unstable radially ~for K.0!. The positive term in the second curly braces represents the energy required to reorient the top’s axis from vertical to the local field direction. This reorientation energy creates the radial potential well at the levitation height when the trapping condition in Eq. ~10! is satisfied.

如果磁矩在偏离中心点时不能自由地指向局部磁场方向,请参见图2 !,则第二个大括号中的项将仅为{-1},并且顶部径向将不稳定,因为K > 0。 第二个大括号中的正项表示将顶部的轴从垂直方向调整为局部场方向所需的能量。 当满足等式中的俘获条件时,这种重新定向能在悬浮高度处很好地产生了磁力井。

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