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[Iacopo]

Ortwin, I didn't understand very much of it, but it seems that you found the correct answer.

[Iacopo]

This is the video with the phase shift of the spindulum Nr. 1.

The top was filmed from above, with the camera centered on the rotation axis of the top, so that it is easy to recognize the direction of tilting of the stem. 
At the right, the footage was edited so to rotate in the opposite direction and with the same speed of the top, so that the view appears approximately like in the frame of reference of the top itself, but with the camera aligned to the rotation axis and not to the geometrical axis of the top;  this makes easy to follow the phase shift.

At the left, you can see the same footage from our normal frame of reference.  You can see the kinematics of the top, the intensity of the wobble, and the RPM. 

At high speed, the stem of the top stays always slightly tilted, (tiny wobble), towards position 180, (phase 180°), as expected; this is the same behaviour we saw in the pendulum.
At some point, the stem moves towards position 90 and then, more and more rapidly, towards position 0.
The wobble becomes intense, especially from phase 90° to phase 0°.
This behaviour too is similar like in the pendulum.

Anyway, at this point, something different happens:
the stem does not stop in position 0, like expected, but it continues to rotate and this rotation accelerates by the time.
Maybe the intense wobble started a precession, I am not sure.
The wobble becomes more intense everytime the stem passes from position 90 to position 0, and less intense from position 270 to 360.
If this is a superposed precession, without it the stem should point approximately towards position 45, like if the transition towards position 0 is not completed yet, but I am not sure about this idea.

[ortwin]

Wow, that is mouthful to  chew on and think about!


First the software to rotate the right video, how does that work? Do you ask it to keep two points at a fixed position relative to th frame? No, I am not sure how that works, I have to watch the video some more times and/or wait for your answer.
 
If that is precession towards the end, does it go the right direction? From what I gather it is not. The spin is initiated clockwise in the spinndulum? The "precession" is also clockwise like in a heavy top? But for a spindulum it should go the other direction, no? And somehow we need to take into account the " spinning camera". ...
I really need some  more time to break it dow into bits I can sort in my mind.
But it really is a great experiment you made Iacopo! It promises further insight and also surprises.

[Iacopo]

First the software to rotate the right video, how does that work? Do you ask it to keep two points at a fixed position relative to th frame?

I use Da Vinci Resolve for editing videos. Rotation can be easily set in the editing page, but only up to 720°, so I had to repeat the task every two revolutions of the top, editing frame by frame.  It took me maybe twenty minutes for to edit that sequence. 

If that is precession towards the end, does it go the right direction? From what I gather it is not. The spin is initiated clockwise in the spinndulum? The "precession" is also clockwise like in a heavy top? But for a spindulum it should go the other direction, no? And somehow we need to take into account the " spinning camera". ...
I really need some  more time to break it dow into bits I can sort in my mind.

I didn't think to the direction but to the increasing speed of that wobble, which is like in precession.
But looking at the direction of the wobble it seems clockwise, like the spin, which in fact is strange.
The top spins clockwise so the precession should be counterclockwise.
I am not sure what to think about.
 

[Iacopo]

This is the new tested spindulum:



The behaviour is similar to that of the first one, but this time the wobble superposed to the unbalance which starts at critical speed, fades away more rapidly, so that it is possible to see that after the phase change the stem stabilizes in position 0°;  this was not evident in the first tested spindulum but now we can see it clearly.  So another piece of the puzzle finds its place.

At high speed, the phase is always 180°.
At the decreasing of the speed, the wobble amplitude increases until reaching the critical speed during which the wobble amplitude is the largest and during which the phase changes from 180° to 90° to 0°.  At low speed the phase is always 0°.  The amplitude of the wobble decreases until fading completely away;  in fact there is not static unbalance in the top, and the couple unbalance, which is related to the centrifugal force, practically does not show anymore at very low speed.   

After the critical speed phase, some intermittent wobble starts: this also tells the existence of a temporary wobble superposed to the unbalance.
It is not clear what is this wobble. Maybe it could be simply the top oscillating at its own natural frequency ??
I should try to see if the natural frequency of the top changes depending on if the top spins or not.

As Ortwin suggested, and for curiosity, I tried to observe the changes of the traced trajectory of the stem of the oscillating spindulum while not spinning, (in the beginning of the video).  These changes are due to the spindulum oscillating a bit more slowly about one horizontal axis, because of the added weights.  The difference is very little, I counted 156 oscillation per minute in one direction and 158 oscillations per minute in the other direction. The added weight decreases the oscillation frequency by 2 oscillations per minute. 

[ortwin]

Do you mean by "critical speed" the resonance frequency you observe while it is spinning?
If so, why is it so far off the natural frequency this time?  It was only  little bit off with the last spindulum, right? Or did I overlook something?

[Iacopo]

Do you mean by "critical speed" the resonance frequency you observe while it is spinning?
If so, why is it so far off the natural frequency this time?  It was only  little bit off with the last spindulum, right? Or did I overlook something?

Yes, the speed when the phase is 90°, and the wobble most intense.
I don't know the reason.  AMI and TMI are larger now, relatively to the CM-tip distance.

[ortwin]

All very interesting and also surprising!
The two frequencies this time are so different that their cause seems unrelated.
But then I still would expect some resonance near the "natural" frequency of the spindulum.
Before thinking any further, it is of course easier for me to just ask you for more data first: What happens to those two frequencies if you change the value of our main suspects (AMI, TMI, H (distance tip - CM) one by one ? What happens in the first spindulum if you do that?
Now, I know it is not easy and probably not even possible to change all those values independently of each other. Probably you could change H from 20 to 10 mm, but then you would have to add some weight to the stem to keep TMI more or less the same without changing AMI too much. Maybe some more thinking could be useful too ........
 
How do you plan to switch over from a spindulum back to a spinning top, the case we are even more interested in?
My intuition is telling me, it would be good to have a "spin thing" where you could change "H" to "- H" while keeping AMI and TMI the same. Not sure if that is exactly possible, but it should be possible to get close to that if you keep the absolute value of H at the low end, below say 5 mm.
Just a relatively spontaneous idea, maybe you are having already better plans for "the switch" or maybe my idea is not one that leads to anything useful.
Edit: Did you measure all those values (TMI, AMI, natural Frequency, weight) independently, or did you determine some of them using a formula from the other values? Just wondering, not that I think the one or the other would be better for anything.
 

[Iacopo]

My intuition is telling me, it would be good to have a "spin thing" where you could change "H" to "- H" while keeping AMI and TMI the same.

Incredible !  You read in my mind.  This is exactly what I planned to do.

Edit: Did you measure all those values (TMI, AMI, natural Frequency, weight) independently, or did you determine some of them using a formula from the other values? Just wondering, not that I think the one or the other would be better for anything.

They are measured indipendently.

I found something interesting that I will post tomorrow about the difference between natural frequency and resonance frequency.

[ortwin]

...

They are measured indipendently.

...

I just checked quickly, roughly calculated the frequency using the other values you gave and the basic formula, and of course it matched. You probably checked that yourself  before as well ......

[ortwin]

......traced trajectory of the stem of the oscillating spindulum while not spinning....

Another idea: if you can make the little upper surface of the stem completely flat and shiny as a mirror but at the same time exactly perpendicular to the spin axis, you would have a nice enlarged trajectory showing at the ceiling  when you point a beam of parallel light to it at some angle from above. Basically the "laser balancing method".

[Iacopo]

I just checked quickly, roughly calculated the frequency using the other values you gave and the basic formula, and of course it matched. You probably checked that yourself  before as well ......

No I didn't. I get the radii of gyration by a bifilar pendulum then I calculated the moments of inertia.
What is this formula ?

[ortwin]

...
What is this formula ?

T = period of your oscillation in seconds
I =  TMI
m = mass of your spindulum
d = distance CM to contact point
 mathematically this leads to an imaginary frequency if "d" becomes negative as in a spinning to as opposed to a spindulum. The appearance of imaginary frequencies is associated with an "unstable equilibrium", which also makes sense in the case of the top: An upright top that is not spinning is in an equilibrium state, only it is not a stable one.

[Iacopo]

Thank you, I didn't know it.  It works.

I thought in a similar way. A normal top cannot oscillate, so there is not resonance, but there is the phase shift like in spindulums, so almost the same dynamics seems still to happen.  Two tops, with same AMI, TMI, and CM-tip distance, but one with external tip and the other with recessed tip, maybe have the phase shift at the same spin speed.  We will see this.

[Iacopo]

INFLUENCE OF THE SPIN SPEED ON THE NATURAL FREQUENCY

I tried to observe what it happens if I make the spindulum Nr. 2 to oscillate while spinning at slow speeds.

The frequency of the oscillation doesn't change significantly at the various spin speeds, but the top spinning makes the oscillation to spin too. So, while in the frame of reference spinning together with the top, the oscillation frequency is relatively stable, in our frame of reference instead, the oscillation frequency appears accellerated by the spinning movement.

Also it can be observed that the rotational speed of the oscillation is slower than that of the top;

I measured these speeds, and I discovered that the ratio between them is constant:


Frame     Spindulum              Oscillation            Ratio
            rotational speed     rotational speed
                 RPM                       RPM

Second       11.2                       3.47                 3.23
Third          42.8                      13.34                3.21
Fourth        87.5                      27.4                  3.19
Fifth          120.4                     37.2                  3.24   


(this is a slow motion, 0.25 x)
(The sixth frame is precession, not oscillation, the stem moves counterclockwise instead of clockwise like in oscillation)

Then, you can imagine what I thought next:

Being the ratio constant, what should be the rotational speed of the oscillation at the frequency of resonance, (227 RPM) ?

227 : 3.2 = 70.9 RPM

And what is the difference between the resonance frequency and the normal natural frequency when the top doesn't spin ?

227 - 157 = 70 RPM.  It's practically the same. Here it is the reason of the difference.

I believe that 157 RPM is the natural frequency when the top is not spinning, and 227 RPM is the natural frequency when the top spins at that same speed. 
That wobble appearing after the resonance phase should be just the top oscillating at its augmented natural frequency.

The natural frequency increases with the spin speed.
Even if I don't understand yet, exactly, how the spin movement makes the oscillation too to spin in its same direction, (but at a lower speed).