Levitron Exam

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A few summers ago (2014) I was browsing in a little shop in Fort Collins, "Science Toy Magic", and saw a demonstration of the Levitron - a small spinning top with a built-in permanent magnet floating a few inches above a large permanent ring-magnet base.
The top would be spun-up by hand on a plastic platform fitted onto the the base, raised up by hand on the platform an inch or so until the top lifted off slightly to float freely in the air right above the base for a minute or two, till it slowed and dropped into a cup on the reverse side of the platform (set back into the base while the top was floating). The shop featured about half a dozen other toys involving floating magnets, but they all incorporated either electronics or a single point of contact (restricting a degree of freedom of movement) to keep an object floating or suspended. I found the Levitron very interesting because it involved no electronics or contact points, just the spinning of the top to maintain its stable floating above the base. So I bought a Levitron, the proprietor Matt Hannifin first ensuring that I could get the top to float.

The Levitron Ultimate package, by Fascinations, includes a pamphlet of instructions and brief theory of operation, small weight washers for the top to compensate for temperature variations, and adjustable leveling feet for the base magnet. 

Shown here are the top, base, and weight washers. The top is too light to spin in the stable volume above the base magnet; the correct weights, fitted onto the top's stem, bring the top into the stable volume. I painted the top's rim black except for a short section, to provide a light pulse to a photo transistor when the spinning top is lit with a light source.

After a week or so of playing with my new toy, I searched on-line for an explanation for its behavior. In my opinion, the best is a 1996 paper "Spin stabilized magnetic levitation" by Simon, Heflinger, and Ridgeway (referred to here as the "S.H.R. Paper"). It consists of a brief history of the toy (invented around 1983), a theoretical treatment that includes the prediction of an upper (and lower) spin rate limit beyond which the region of stable floating disappears, an experimental setup for verifying the prediction, and comparisons of the commercial version of the top with some alternate versions built by the authors.
The experimental setup suggested some fun with electronics to be had with my toy. It would be interesting to electronically drive the top to its upper stable-floating spin limit to find out how it compares to the S.H.R. Paper's results; and maybe to measure the precession rate, which is necessary for floating and which decreases as spin increases towards the upper stable-floating spin limit. For this top, the drive electronics would not be used to create the stable-floating state, but just to vary the spin rate of the top within the stable region determined by the magnetic fields of the top and base.

To start with, I shamelessly borrowed from the experimental setup described in the S.H.R. Paper:

- Two 10" diameter coils wound with 70 turns (and a few feet extra for connection) of 26 ga. enameled wire for about 8 ohms impedance each. The coils are set up vertically each side of the Levitron base about 8" apart in a Helmholtz-coil-like arrangement with central axis through about where the top would float. The coils were held in a frame built of Legos which surrounded the base magnet. This setup was used to determine if the top could be driven at all.

- To avoid possibly destroying the family sound system, I bought a 20 amp stereo amplifier (Lepai, from Amazon) for about $20 to drive the coils. I expected to be using the stereo feature to drive a "rotating" magnetic field at some point.
- A shareware program variable "oscillator" driving my computer's sound card, to provide the drive signal via the amplifier to the coils. A very nice one which has met all my experimental needs so far has been the Test Tone Generator (version 4.33) by Timo Esser. It produces 9 wave function types (including sine, which I used) of about 20Hz-20kHz, constant or sweep frequency, further amplitude modulation of the wave functions, and independent control of the stereo channels. Also, a shareware PWM (Pulse Width Modulation) Generator application from the same source was used for some experiments with pulsed drive signals

- A photo transistor (two leads) with a 12 volt supply in series with 3.3k ohm resistor. A small LED flashlight illuminates the spinning top's rim, which is painted black except for a small section, providing a light "blip" to the transistor with each turn. The photo transistor's output is then monitored for spin period. I rebuilt the frame to support the base magnet and move the coils closer to about 6" apart, and added Lego supports for the flashlight and a plastic tube containing the photo transistor. I found spin rate monitoring was needed when it became apparent that the stable spin rate was not synchronous with the coil drive frequency.

- An oscilloscope for monitoring driving frequency and amplitude, and spin frequency. My old Heathkit Dual trace scope was just right.
I started experimenting by trying to sync the spinning top to the driving signal, presumably by means of the top's "residual transverse magnetization" mentioned in the S.H.R. Paper. Using the Helmholtz coil arrangement and the same sinusoidal signal to both coils, I slowly increased the signal amplitude at a constant low frequency, till the spinning top would start to give off a soft shaking sound. Often the top would then continue to spin well past its unaided stop time of one to two minutes - many minutes or hours (My record run was 18 hours of continuous spin in an air-conditioned bedroom maintained at about 70F degrees.) However, I was having trouble maintaining spin at driving frequencies of around 30Hz or lower, much higher than the low spin limit reached in the paper. A drive frequency of 40Hz worked quite well for long spin times.

Then I hooked up the photo transistor sensor and found that the stable spin frequency was about 3/4 (but not exactly) of the drive frequency at 40Hz, and a higher fraction still at lower drive frequencies. The top seemed to be acting like an induction motor rather than a synchronous one. Sure enough, later the S.H.R. Paper mentions that the magnets used then were ceramic (the significance of which I had missed), which are non-conductive, while my top's magnet is a very conductive neodymium alloy washer. An online search turned up other mentions of earlier Levitrons using ceramic magnets in tops.

Deciding that I didn't need the drive coils' field to be as uniform as provided by the Helmholtz configuration, and wanting to provide a "rotating" drive field if needed, I rearranged the coils to be vertically at right angles to each other and enclosing the Levitron base (again supported by a Lego platform) so that the top would float at the centers of the two coils. It soon became clear that a rotating field just wasn't needed, the single phase signal being sufficient to drive the already manually spun-up ("started")  induction top "motor". The two coils (each on its own stereo amplifier channel) were thus driven at same amplitude from a sinusoidal monaural source with no phase difference. At the center, the resulting magnetic field is directed along one (of two possible, depending on coil winding direction) horizontal axis midway (45 degrees) between the axes of the two coils, and varies in strength sinusoidally back and forth along this axis at the source frequency, with an amplitude 1.41 times that of either coil alone.

An added benefit of the right-angle coils configuration turned out to be that it was easier to get my hands in to start the top and raise its platform without moving the flashlight & photo transistor towers out of the way.
Here is an article on single phase induction motors that may help in understanding how the rotating top can be induced :-) to continue turning by a non-rotating (but alternating) magnetic field:  "Why Single Phase Induction Motor is not Self Starting?" (and why it continues to run once started).
The chart  shows some top spin measurements with coils at right angles, with single phase non-rotating magnetic field drive signals.

I was trying to discover the maximum spin rate for the top - when it "flies off" over the top and off to the side of the stable volume, and the minimum rate - when it drops through the bottom of the stable volume into the base ring (where it's captured by the plastic cup on the reverse side of the plastic launch platform, to prevent damage or loss of scattered parts).
For comparison, my limiting spin rates from this chart are somewhat higher than those experimental rates for Levitron base and top shown in Table I of the S.H.R. Paper (which I assume was done at about standard room temperature, 68F degrees, close to my 70F case):
Case:                 Upper limit (RPS):      Lower Limit (RPS):
My rate               46.8 (70F)              26.6 (70F)
   "                                          22.9 (60F no drive)
S.H.R. Paper rate     40.4 (254 rad/s)        18.1 (114 rad/s)
So what other Table I items could I compare my Levitron with, using inexpensive calibrated equipment on hand or easily obtainable?
Well, there is the S.H.R. Paper's:
- top mass = 0.02135 kg
- rotational inertia I = 2.2x10^-6 kg m^2 (directly measured)
- transverse inertia It= 1.32x10^-6 kg m^2 (directly measured)
- It/I = .60 (calculated)
- reff = 0.0102 m (calculated)

Other quantities related to magnetic field strength were beyond my measurement means.

For mass (from weight) measurements I used an AWS-100 Digital Scale (about $10.00 less shipping from Amazon) with a 100 gram capacity and "Hundredth of a gram accuracy", along with (separately) a 100g calibration mass. To avoid unduly affecting the scale readings by the top's magnet, I weighed the top using two light supports of different heights (a tall plastic pill bottle, and an even taller TP roll tube), subtracting the support weight to get that of the top (the scale has a function to do this), and ensuring that both top measurements were identical.
For the dimensional measurements of the top's magnetic rotor and weights, I used an "iGaging EZ-Cal IP54 Digital Caliper (about $27.00 less shipping from Amazon), 6" range and "0.001 accuracy". To measure the rotor's hole diameter (filled with translucent plastic stem and molded around it), I used strong back-lighting to view the hole through the top's point end and the calipers to frame the hole's diameter. For most measurements of the top's plastic stem, I used a transparent ruler with millimeter scale and near-sighted eyeballs.
A typical weight washer mass & dimension:
- Orange plastic: mass= 0.42 g, O.D.= 25.55 mm, I.D.= 6.5 mm, thick= 0.85 mm
To calculate Top Moments of Inertia: Rotational (about the top's spin axis), and Transverse (about horizontal axis through top's center of mass as it flips from vertically upright to upside-down) types. The calculation is done by breaking the stem into its component shapes and getting the moments each shape separately, then adding the moments of each type together to get a totals for the rotational and transverse moments of the stem. These are added to the moments obtained for the magnetic rotor piece to get the total two moments for the top (without weights). The two moment types are also calculated for the weight washers, so the rotational and transverse moments for a weighted top can be obtained. For the Transverse moments, I made the assumption that all parts "flipped" about their centers (close to true), except the stem top part which is flipped about its base end, and the pivot point hemisphere, which is flipped about its base.
The moment formulas I used are as follows, obtained from Wikipedia topic "Lists of moments of inertia" and checked for understanding against my copy of Physics for Students of Science and Engineering, Halliday and Resnick. I coded the formulas into a VB application, knowing that I would be doing the calculations more than once...

- For the weights and top magnetic rotor (look like washers):
Thick walled cylindrical tube, open at ends: mass m, height h, inner radius r, outer radius R:
Irot= m/2 * (r^2 + R^2)  -- about axis through tube
Itrans = m/12 * (3 * (r^2 + R^2) + h^2)  -- about axis through center of tube and perpendicular to it
- For the top's stem:
I trans_at_end = Itrans + m *(h/2)^2  -- about axis at end of tube , through its central axis and perpendicular to it
- to make the tube (of mass m) solid, set inner radius  r=0.
- to make the tube (of mass m) a thin shell of the outer radius, set inner radius r to the outer radius: r = R. (not used here)
- For the hemispherical bottom "point" of top:
Solid ball, mass M, radius r:
Iball = 2/5 * M* r^2 (not used here)
Solid hemisphere, mass m, radius r
- Ihemi_rot = 2/5 * m * r^2  -- about axis through center of base, perpendicular to base
- Ihemi_trans = Ihemi_rot -- about axis through center of base and along base

I wrote a simple VB program to add up the mass, and rotational and transverse moments of inertia for any configuration of top plus weights, by selecting the items wanted via check boxes. (Later I added the capability of determining the configuration needed to float the top based on temperature, as described on page Levitron Temperature Sensitivity Details.)

A screen shot of the program is presented here because the resulting hard-coded mass and and moments of inertia values for the components are all displayed, previously calculated using the formulas above. (the top's values are for the top rotor plus its stem.)

Assuming the S.H.R. Paper's Table I Levitron top values are for a weighted top to float at about a comfortable lab environment (room) temperature (~68F), I have:
                       mass: g  rot MI:           trans MI:        It/I
                                (x10^-6 kg m^2)   (x10^-6 kg m^2)
My unweighted top:     16.43    1.84              1.04             0.56

S.H.R. top (~68F deg): 21.35    2.20              1.32             0.60
My top for 68F deg:    19.93    2.12              1.18             0.56
My top for 27F deg:    21.35    2.21              1.23             0.56
Weights need to be added to the top to allow it to float at lower temperatures, so about 3.5 grams needs to be added to my top for it to float at 68F, and another 1.72 grams for it to float at 27F. The above table includes these cases, with the 27F case showing comparable rotational MI for my weighted top and the S.H.R. Paper's top (assumed weighted for ~68F)  for the same total masses.
The weight needed to float the Levitron top at different temperatures is investigated further on page Levitron Temperature Sensitivity Details.
The S.H.R. Paper's (written in 1996) Levitron base magnet was square:
"In 1994 Bill Hones [of Fascinations] and his father applied for a patent on a levitating top that used a square permanent magnet base, which was issued in 1995. The Levitron, made by Fascinations, has a square base magnet with a region of weaker or null magnetization in the center." (S.H.R, p287)

My Levitron base certainly appeared to contain a circular (ring) base magnet...
Interested that there were no construction surprises in the Levitron base, I took a look inside, turning it over and removing three small retaining screws, three threaded balance feet, and then the base plate. There were no surprises: The ceramic ring magnet is glued to the underside top of the base, the base and feet are made of plastic, and there are no other magnetic materials  besides the retaining screws and balance feet threads.
The magnet dimensions are:
- 110 mm outside diameter
- 58 mm inside diameter
- 20 mm thick
- If a flat plate were placed on the upright base, about 4 mm would separate the glued magnet face from the underside of the plate.

For the top to float above the base, the base feet are adjusted so that the magnetic field direction above the base is vertical very close to the central axis of the base (and its contained ring magnet). This is typically done by trial and error, using the top to incrementally approach the balance point by seeing which way it falls away when trying to float it, and making compensating adjustments to the base feet - a time consuming process that needs to be repeated whenever the base is moved to a new location.
To shorten this process, I used a circular "bullseye" bubble level on a CD case lid placed on the base, but quickly found out that due to construction and ring magnet imperfections, a horizontal base does not guarantee a vertical field direction close enough to require only a few touch-up adjustments to get the top to float stably.

The photo shows the level bubble position after the base has been adjusted properly to float the top.

I built up a shim on one end of the CD case lid so that when the lid is positioned with the shim side down and  opposite the high point on the base (marked with white tape in photo),the bubble would show level when base adjustments leave the magnetic field close to vertical. Measuring the height of the shim and the distance from it to the lip at the mark gives a correction angle of about 1.15 degrees for my base.