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[Russpin]

Getting closer to the transition point. At rcm = 0.436 the top is still a little on the leaning toward the point mass side. Figure 3 shows the time histories of theta, psi, phi dot ( precession rate) and w3 ( angular velocity along the z axis aka 'spin'). The simulation was run for one hundred seconds.  At time zero theta is very near zero ( can't be exactly zero because the equations have a divide by sin theta) theta increases to about 35 degrees and the top stays there a while and then starts to come back towards vertical. It's also seen that the precession rate is about equal to the initial spin rate. This is very different than the precession rate for a normal balanced top where the precession rate is much less than the spin rate. This high precession rate causes psi to change slowly with time which is also very different from the normal balanced top where psi changes very quickly with time. The next post shows plots for a normal balanced top.

[Russpin]

The simulation was run with the added point mass set to zero (mp = 0.0) and the initial theta set to 20 degrees. Initial spin (w3_0) was set to 3. rad/s. This is the normal precession and nutation of a balanced top.

[ta0]

Getting closer to the transition point. At rcm = 0.436 the top is still a little on the leaning toward the point mass side. Figure 3 shows the time histories of theta, psi, phi dot ( precession rate) and w3 ( angular velocity along the z axis aka 'spin'). The simulation was run for one hundred seconds.  At time zero theta is very near zero ( can't be exactly zero because the equations have a divide by sin theta) theta increases to about 35 degrees and the top stays there a while and then starts to come back towards vertical. It's also seen that the precession rate is about equal to the initial spin rate. This is very different than the precession rate for a normal balanced top where the precession rate is much less than the spin rate. This high precession rate causes psi to change slowly with time which is also very different from the normal balanced top where psi changes very quickly with time. The next post shows plots for a normal balanced top.

Nice!
It would seem that symmetric peaks are developing to both sides of the heavy side. It is tempting to guess that at the transition height there are peaks at +/-90 degrees to the heaviest point. I wonder if carefully using the paint brush balance method in this case would give marks with gaps on the heaviest and lightest sides.

I found it confusing at first to think about phi and psi in this case, because phi which measures precession for a regular top becomes mostly a measure of the spin close to the transition. It helped me to think about the rolling cones of the inertial precession case. The transition happens when the static cone becomes very narrow. At that point the moving cone will circle many times (measured by phi) while the center of mass only slowly turns towards the center (measured by psi).

You are doing great work!

[Iacopo]

It is tempting to guess that at the transition height there are peaks at +/-90 degrees to the heaviest point.

I too thought to this as a possibility, also if practical results from me and Alan do not seem to point in this direction.
But more tests with different tops would be needed.

[Russpin]

To facilitate an easier comparison with Iacopo's plot showing the effect of height of center of mass and spin speed (400 and 900 RPM). I put in Iacopo's top parameters into the simulation:

Mass = 0.0429 kg
Diameter = 0.073 m
Thickness = 0.0128 m (this is an estimate based on the photo)
g = 9.8 m/s^2

Mass of unbalance point mass = 0.000429 kg ( 1 percent of top's mass, this is an assumption)

The following plots show that for an rcm of 18 mm the top leans directly away from the added point mass (psi = 90 degrees) at both 400 and 900 RPM. For an rcm of 14 mm the top leans directly towards the added point mass at both 400 and 900 RPM.
The units shown in the plot titles are:
I1, I3 in kg*m^2
w3_0 in rad/s ( 400 RPM = 41.888 rad/s,  900 RPM = 94.248 rad/s)
rcm in m

The next post shows the results for the transition height of 16 mm.

[Russpin]

The following plots show that at the transition height of rcm = 16 mm, the top leans directly away from the added point mass at  400 RPM and leans directly towards the added point mass at 900 RPM. This does not match Iacopo's plot which shows intermediate angles at these points.

[ta0]

Again, beautiful results (and the graphs are beautiful in themselves)! 

The RPM dependence at the transition point, and only there, was a surprise.

Perhaps if you explore in more detail around the transition point (intermediate spin rates?) you will find intermediate angles. 

[Iacopo]

The following plots show that at the transition height of rcm = 16 mm, the top leans directly away from the added point mass at  400 RPM and leans directly towards the added point mass at 900 RPM. This does not match Iacopo's plot which shows intermediate angles at these points.

In fact in my top I had a more gradual transition at the changing of speed.


 
The mass of the unbalance point in my top was about 1/500 of the mass of the top, could this be the reason ?
The thickness of the flywheel is mm 13.7.
I don't know if it matters, but there are three brass grub screws in the top, each one 3.6 grams, mm 12 long, staying near the outside of the top, (total mass of the top, screws included, grams 42.9).

Anyway I find interesting that there is a similar result for the transition height of CM in theory and in practice.

Also I find interesting that the influence of speed is in the sense of making the top to lean towards the added mass at higher speed and away from it at slower speed, which is the same effect I have seen in my top, (and Alan too has seen the same in his top).

Russpin, you have made a great job with these plots !  I couldn't imagine that there is a software able to predict the behaviour of spinning tops so accurately.

It would be nice to know if you can find the intermediate angles...



 

[Aerobie]

These simulations are indeed impressive. 

I have an observation on balance, which I'd like to share. 

As I've written in prior posts, I record decay rates as the top slows down.  The decay rates are not perfectly reproducible, because I apply a very thin haze of oil to the mirror (base) which influences decay rates.  However, the amount of lube is pretty stable from one spin to the next.

Today I had two sequential spins with the same top, before and after balancing.  The decay rates above 120 RPM were about the same.  But below 120 RPM the decay (loss of RPM per second) reduced to about half after balance. 

I hasten to explain that most tops have already toppled before getting down to 120 RPM, but I've developed designs that reach zero RPM still standing.  The newest top has a decay below 120 RPM of only 1/4 RPM per second after improved balance.  So over five minutes add to the spin time between 120 and zero RPM.  Incidentally, although the tops remain standing at zero RPM, I can't place the top on the mirror so it remains standing.  Although that would seem possible with perfectly vertical placement.

So in summary; improved balance substantially reduced decay rate.  I theorize that before improving balance, the tip was "scrubbing" due to slight wobble which dissipated energy.  The benefit of balance was confined to low RPM, because that's the range of greatest wobble.

Alan

[Russpin]

Again, beautiful results (and the graphs are beautiful in themselves)! 

The RPM dependence at the transition point, and only there, was a surprise.

Perhaps if you explore in more detail around the transition point (intermediate spin rates?) you will find intermediate angles. 

Thanks Ta0, for all your insight and encouragement!

The mass of the unbalance point in my top was about 1/500 of the mass of the top, could this be the reason ?
The thickness of the flywheel is mm 13.7.
I don't know if it matters, but there are three brass grub screws in the top, each one 3.6 grams, mm 12 long, staying near the outside of the top.

I will try some of these changes to see if it makes a difference.

Russpin, you have made a great job with these plots !  I couldn't imagine that there is a software able to predict the behaviour of spinning tops so accurately.

It would be nice to know if you can find the intermediate angles...

Thanks  Iacopo, it was your work on unbalanced top behavior that inspired me to write this simulation.
I'm pleased that the simulation is predicting things as well as it does. However as they say “the devil is in the details” and the gradual transition and intermediate angles are two things that need to be resolved.