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[ta0]

The diameter of the ceramic ball is mm 19.05, and the angle of tilting was 1.58°
The ball top made 115.5 spin revolutions in 1.51 seconds, (at 4589 RPM).
The calculated traveled distance is mm 190.5
The length of the track is mm 214.8
There is a bit of uncertainty about the angle of tilting, I believe that my measurement is not wrong by more than 0.2°, . . .

Obviously, there cannot be negative (forward) slipping created by just the rotation of the ball.
But if we assume no slipping and back-calculate the angle necessary, it gives 1.78°, just the maximum error of 0.2° that you estimated above the measured angle.
So, I'm inclined to think there was no slipping. A larger angle may give a more precise measurement.

[Jeremy McCreary]

Iacopo: Are you saying that the line actually traced out by your test top is =longer= than the line it would have traced without tip slip -- i.e., with pure rolling throughout?

[Iacopo]

Iacopo: Are you saying that the line actually traced out by your test top is =longer= than the line it would have traced without tip slip -- i.e., with pure rolling throughout?

Yes.
But probably Ta0 is right, so maybe there was not slipping.
Today I will try again, with a less narrow angle of tilting of the ball top.

[Iacopo]

Today I had again negative slipping.

The ball top traveled the examined track, (218 mm), in 1.15 seconds, by 38.7 spin revolutions, at 2019 RPM (average).
Angles of tilting have been 5.9° at the start, 5.0° at half timing, and 4.45° at the end, (average 5.13°).
Calculated traveled distance, mm 207
Length of the track, mm 218
Percentage of negative slipping, (forward): 5.3 %

From the video, (checking it frame by frame), I noticed that the traslational speed of the top at the beginning of the examined track was slightly slower than at the end, so it seems that the top was still in the traslational acceleration phase at the beginning.
I will have to prepare a longer acceleration segment for the top, to avoid this problem in the next measurements.
The faster rising in the first half of the timing could be associated with this residual acceleration.
But, if so, without this acceleration, the percentage of negative slipping would be even higher than 5.3 %.

This is so confusing, but I can't find anything I am doing wrong. 
I checked the camera and the screen for distortions of the image, (I measure the angles of tilting of the top in the screen of my PC), but the image is very good and I didn't notice any distortions.


 

[ta0]

From the video, (checking it frame by frame), I noticed that the traslational speed of the top at the beginning of the examined track was slightly slower than at the end, so it seems that the top was still in the traslational acceleration phase at the beginning.

Mm, if it's not slipping, the translation speed at the beginning should be higher than at the end as the axis is tilted more! 

I believe you are recording this a high speed, so there is no error on the number of revolutions. The track distance you can measure precisely. The measurement of the angle would be the most likely source of error. Did you recheck the diameter of the ball?

[Jeremy McCreary]

Today I had again negative slipping.
This is so confusing, but I can't find anything I am doing wrong. 

I'm stumped, too. You and ta0 have identified the main sources of error but no clear suspects. Agree with ta0 that we've seen no definite slip signal in your current test top.

I also have tops with a strong tendency to travel in roughly staight lines. Some of these definitely slip right after launch (think wheelspin in a dragster). Then the tips "hook up" and roll thereafter.

Top travel with tip slip is kinematically complex, and I've had a hard time thinking about it in non-mathematical terms.  But thanks to your experiments with straight-line travel, I'm now approaching the kinematics by starting with this simple precession-free case -- a well-understood basic problem in vehicular dynamics with imperfect traction.

Then maybe I'll be ready to move up to curved tip traces with slip in any direction.

[Iacopo]

The diameter of the ball is mm 19.05.
The video is at 50 fps, which is sufficient for to count the number of the spin revolutions.

I believe that the highest speed was not at the end of the track but a little distance after the start.
So there was translational acceleration for a short distance after the start, and then deceleration until the end.
The rise of the top was rapid, from 5.9° to 4.45° in little more than one second;
maybe the top started to brake, because of this, and if it braked slipping, this could explain the slipping forward.

I will repeat the test taking the data of the ball top spinning in two consecutive examined segments, so it is possible to see how the behaviour of the top changes by the time. And of course I will be more careful to have the top at the start of the examined track already at full traslational speed, and not in acceleration.

[Iacopo]

Today I tried the ball top without grease on the glass spinning surface, to have better grip for the top.

I found a very good goniometer in DaVinci Resolve, it is more practical to use than a real goniometer in this situation, and also more reliable.  Now I think I can take angle measurements with error no more than 0.1°.

I divided the examined track in two segments, so I have the data for the first and the second half of the track separately:

                              First half                Second half
Length of track        mm 247                 mm 242.5
Time                       1.075 sec              1.120 sec
Traslational speed    229.8 mm/sec       216.5 mm/sec
RPM                        1494                     1484
Spin revolutions       26.77                    27.70
Angle of tilting         From 8.8° to 8.7°   From 8.7° to 7.75°
Average                  8.75°                     8.22°
No slip length          mm 244                 mm 234

Percentage of slipping, (forward):  1.2 % in the first half of the track
                                                  3.6 % in the second half of the track

These data are more "normal".  In the first half of the track maybe there was no slipping, because 1.2% is low and could be due to measurement errors.  But 3.6% is not so low, so maybe there was really some traslational forward slipping in the second half of the track, maybe because the top was braking, (the traslational speed was lower in the second half).

I am not sure about the reason of some differences between the first and the second parts of the track:
braking seemed stronger in the second half.
Above all, the rising was much more rapid in the second half, (and it's real, I'm sure there are no errors here);  I suspect that this could be due to imperfect planarity of the glass spinning surface, which is lower by about 1 mm in its center, so that the top goes downhill at the start and uphill in the end.  Tomorrow I will add some support in the center to eliminate the defect.
I will try again with a thin layer of grease on the glass. 

[Iacopo]

I have improved the planarity of the glass spinning surface.

I repeated the test with the ball top, with a thin layer of grease on the spinnig surface.

                               First half            Second half     
Length of track         242 mm             243.3 mm
Time                        0.658 sec           0.766 sec
Translational speed   368 mm/sec       318 mm/sec
RPM                         2545                  2522
Spin revolutions        27.9                   32.2
Angle of tilting          start 9.1°           start 8.1°
                               end  8.1°           end  6.9°
                               average 8.6°      average 7.5°
Calculated distance
traveled by the ball   249.6 mm          251.4 mm


For the first time I found slight positive slipping in the ball top, in spite of the relatively strong translational deceleration, and in spite of the lubricated spinning surface, which maybe could be expected to produce negative slipping (forward), but this did not happen.
And, if this did not happen with these favourable conditions, all the more reason it should not happen without them.
I am starting to think seriously that there is no slipping in the ball top, and that the forward slipping found in the last days was due to inaccuracy.  I will continue with the tests, to be surer.

The rise of the ball top was very fast, as usual. The grease, increasing the rolling resistance, makes the rise even faster.
   

[ta0]

Is the disc perfectly centered on the ball so their centers of mass coincide? In that case it should rest at whatever angle you leave it and it shouldn't precess while spinning. So the fact it stands up is not due the same mechanism that makes at top stand up and sleep.

[Iacopo]

Is the disc perfectly centered on the ball so their centers of mass coincide? In that case it should rest at whatever angle you leave it and it shouldn't precess while spinning. So the fact it stands up is not due the same mechanism that makes at top stand up and sleep.

I checked it, the flywheel is very slightly below the center of the ball, so there is a very weak tendency to self right.
Trajectories are straight, or very near to the straight line.
I will correct this, even if I don't think that this will change the behaviour of the top significantly.

I see two torques acting on the rise of the ball top;  the translational deceleration, which tends to make the top to fall forwards, and has the effect to make the top to tilt down. And rolling resistance, which makes the top to rise.  I believe that the second is stronger than the first, so the top rises.

Or, at least, I don't have alternative explanations.  If rolling resistance was weak, what else could make the top rise, and so fast ?
 

[Iacopo]

I improved a tiny bit the precision of the tests;
I painted the flywheel of the ball top black, for a slightly more accurate measuring of its angle of tilting, on the white background, (reflections on the brass didn't help).
I centered more precisely the height of the flywheel, so to eliminate whatever residual precession.
I realized that another tiny source of error comes from the camcorder; for these sequences I set the shutter speed at 1/800 second, anyway the image is not recorded all at once, but from above to below, in about 1/80 second.  I use a mirror above the spinning top to see it from above, for to know the start and end points of the examined tracks;  since I have to move the mirror while the top spins, the image of the top is not always at the same height in the frame;  I noticed the problem because, in the video, the mark on the top in the side view appears in a different position than in the view from above, (you can see this if you pause the video).
Now I consider all of this in my calculations.

It may seem boring to repeat continuously the same test, but, by refining the method, and the calculations, the results become more reliable:

I repeated the test considering the margins of maximum error, (minimum and maximum values);


                           First half                      Second half
                           of the track                  of the track

Measured data:

Track length         mm 239.6-240.0        mm 239.4-239.8
Time                    0.500-0.504 sec           0.532-0.536 sec
Spin revolutions    33.53-33.63                35.52-35.62
Angle of tilting      start      7.0°-7.3°        start 6.5°-6.7°*
                           end       6.5-6.7°*        end 5.9°-6.2°
                           average 6.75°-7.0°      average 6.2°-6.45°

Calculated data:

Translational speed  475.4-480.0 mm/sec   446.6-450.8 mm/sec
RPM                       4008-4019                  3991-4002
Distance traveled
by the ball top        mm 235.7-245.2     mm 229.5-239.4

Percentage of slipping   -1.7%,+2.2%              -4.4%,+0.0%   

The results are compatible with the top not slipping.

* The image of the top on the frame is better focused at halfway across the track, so here the margin of error is reduced from 0.3° to 0.2°.

[ta0]

That's a very good experimental execution. Top notch! 

[Jeremy McCreary]

An impressive tightening of your accuracy and precision. The video really helped me understand your measurements.

Makes sense to me that your particular setup involves little or no slip -- forward, backward, or sideways.

Trying now to figure out what these results tell us about self-righting in general -- especially with other combinations of CM-contact distance H and tip radius of curvature at the patch R.

In your latest test top, I gather that H = R . In my self-righting tops, and in most others I've seen,  H > R -- often by a good margin.

Some of my lower-mass self-righting tops also slip (skid) sideways for a time when launched in certain ways. But best to save that complication for later discussions.

To understand self-righting in real tops, we'll eventually have to understand the extent to which gyroscopic effects shape the curved tip traces we usually observe. In particular, I'd like to know who's really steering in a given trace -- the contact patch or gyroscopic precession. What conditions favor one steering influence over the other?

[Iacopo]

That's a very good experimental execution. Top notch! 

Thank you !