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[Iacopo]

I resume this part, before to go on.

Spindulums can oscillate while spinning. When the speed of the oscillation matches that of the spin, there is resonance, and the unbalanced spindulum wobbles a lot at that point.

The nature of the oscillation turned out to be complex;
the speed of the oscillation is not constant, but increases with speed, partly because of the rotation of the oscillation plane, and partly because the frequency of the oscillation itself appears to increase with speed.





The rotation of the oscillation plane is clearly linked to nutation, because it has always its same direction, and also its speed is always proportional to that of the theoretical nutation.
So, at first, I believed that the speed of the rotation of the oscillation plane is nutation.

Anyway, then, I realized that this cannot be correct;
the spindulum Nr. 4 has the tip very near to the center of mass, (only 0.8 mm), so it cannot oscillate rapidly.  If the tip was exactly at the center of mass, the spindulum could not oscillate at all; in which case, the wobble we are observing would be entirely nutation, without any component of oscillation.
I feel sure that with this very little distance between the tip and the center of mass, the behaviour can't be very different, and that, at high speed, the wobble must be almost entirely nutation.
In the graphic of the spindulum Nr. 4 above instead, the speed of the circular oscillation increases constantly with the spin speed, remaining always faster than the presumed nutation speed, (the speed of the oscillation plane).
This seems absurd to me.

The speed of the oscillation plane is always about half than that of the theoretical nutation.
The explanation seems to work better to me, if I accept that the nutation really happens at the speed it should have, (which is twice that of the plane of oscillation).

The wobble causing the resonance would be composed by oscillation and nutation.
When the spindulum does not spin, there would be only oscillation.
When the spindulum spins at low speed, the dominant component would be still oscillation.
At the increasing of the spin speed, the nutation would become more and more dominant.
The instant when the nutation would become larger than the oscillation would happen with the spin speed at about 25 RPM for the spindulum Nr. 4 and about 105 RPM for the other one.
   

[Iacopo]

AN INTUITIVE EXPLANATION OF THE PHASE CHANGE IN SPINDULUMS

I AM NOT SURE THAT THE FOLLOWING IDEA IS CORRECT ! 
EDIT: I SEE NOW THAT ONE OF THE CONCLUSIONS OF THIS CONCEPTION IS CERTAINLY WRONG.
I AM LEAVING THIS POST HERE, BUT THE EXPLAINED IDEAS ARE NOT CORRECT.



It is possible to think to the phase change in terms of an increasing delay between the driving force, (the spin), and the oscillation, at the increasing of the spin speed and of the TMI_tip/AMI ratio. With unfavourable conditions, the oscillation can't keep pace with the spin and a delay develops until 180°. 

The next is a simpler and more intuitive explanation:

Below, there is the circular oscillation of a spindulum which is not spinning.
The rotation axis is vertical and passes through the tip.  The center of mass rotates about the vertical rotation axis, so there is a centrifugal force pulling the center of mass outwards, (orange arrow).
At the same time, the center of mass is pulled downwards by the force of gravity, (fuchsia arrow).
The resultant force, (red arrow), is aligned with the spindulum axis passing through the tip and the center of mass;
these are the aequilibrium conditions of this motion.



Below, the spindulum is unbalanced, (there is an added mass at one side of the flywheel), and it is spinning, (not oscillating).
The spin axis is vertical and passes through the tip.
While spinning, the added weight stays always in the higher part of the flywheel.

In this case, the centrifugal force and gravity pull the center of mass in direction of the red arrow, but this time the red arrow is not aligned to the tip, but to a point that stays below the tip.
This causes a torque acting on the spin axis that, pivoting on the tip, makes the spindulum to spin staying tilted towards the light side, as you see in the drawing.



In this other case, the unbalanced spindulum spins staying tilted towards the heavy side.
The centrifugal force and gravity pull the center of mass in direction of the red arrow;
the red arrow is not aligned to the tip but to a point above it, this causes the torque that, pivoting on the tip, makes the spindulum to spin staying tilted towards its heavy side.



In this way it is not difficult to guess the conditions that favour the unbalanced spindulum to stays tilted towards the light or towards the heavy side.

With a lower spin speed, the centrifugal force is weaker, (you can imagine the orange arrow in the drawing to be shorter), so that the resultant force, (red arrow), becomes tilted more vertically, and, passing through the center of mass, becomes aligned to a point which is higher in the spin axis.
This means that there can be a phase shift, due to the decreasing centrifugal force.
The unbalanced spindulum can stay tilted towards the light side at high speed and then towards the heavy side at low speed.

The relative position of the center of mass and of the tip are important too.
The position of the tip on the center of mass being high, favours the unbalanced spindulum to spin staying tilted towards the light side.

EDIT: THIS IS NOT WHAT IT CAN BE OBSERVED IN REAL SPINDULUMS: A GREATER CM-TIP DISTANCE FAVOURS THE UNBALANCED SPINDULUM TO SPIN STAYING TILTED TOWARDS THE HEAVY SIDE, NOT THE LIGHT SIDE.

The angles of tilting in the drawings are exaggerated for clarity.

[Iacopo]

I tested the first spinning top.

Unexpected discovery: not only there is phase shift, which I knew, but also there is resonance, like in spindulums !
The wobble becomes more and more intense, until the 90° phase, and then, after the resonance, a secondary wobble appears, superposed to the unbalance wobble.
The secondary wobble, (I don't know yet what is it), interacts with the unbalance wobble causing an intermittent wobble.
After the resonance, the wobble becomes less and less intense, in spite of the lower spin speed.

At speed higher than that of resonance the top stays in phase 0°, and at speed lower than that of resonance the top stays in phase 180°.
This behaviour is opposite to that of the spindulums, which stay in phase 0° at low speed and phase 180° at high speed.

During the resonance, the two wobbles have about the same speed, so the secondary wobble in the beginning is not visible.
After a little time, the two wobbles start to have a slightly different speed and this causes the visible intermittent wobble.
The stem starts to move from position 180° to position 0°, (spindulums), or from position 0° to 180°, (spinning tops).  In the first case the secondary wobble, (oscillation + nutation), has become faster than the spin speed.
In the second case the secondary wobble has become slower than the spin speed.

This is the video showing the resonance and the phase shift of the spinning top:

https://youtu.be/Y_y3rJ9ddTg

These are its data, which can be compared to those of the spindulums.
I will add the data of some more tops in the next days.


 

[Jeremy McCreary]

Very interesting observations, Iacopo!

[Iacopo]

Very interesting observations, Iacopo!

Thank you, Jeremy.

I didn't expect to see resonance, because spinning tops cannot oscillate like spindulums.
So, what is that wobble started by the resonance ?
It seems not precession, nor the regular nutation. 

I add new data, (the top D2 is the D1 with a bit longer tip).

[Jeremy McCreary]

Very interesting observations, Iacopo!

I didn't expect to see resonance, because spinning tops cannot oscillate like spindulums.
So, what is that wobble started by the resonance ?
It seems not precession, nor the regular nutation. 

That wobble is probably whirl — especially if it had the same angular frequency as the spin.

Whirl is an inertial response to rotating unbalance. It's not a gyroscopic effect per se, but gyroscopic effects can alter the speeds at which resonance appears. Unlike precession and nutation, the whirl rate is almost always 1:1 with spin rate. If it were not, the paintbrush method would never work.

[Iacopo]

That wobble is probably whirl — especially if it had the same angular frequency as the spin.

Whirl is an inertial response to rotating unbalance. It's not a gyroscopic effect per se, but gyroscopic effects can alter the speeds at which resonance appears. Unlike precession and nutation, the whirl rate is almost always 1:1 with spin rate. If it were not, the paintbrush method would never work.

That wobble has the same angular frequency as the spin for a while, during the resonance.
Then the frequency of the wobble decreases a bit more rapidly than that of the spin, so the resonance ends and the generated second wobble becomes visible in the form of intermittent wobble of the top.

I have to think to the nature of this whirl.  I have collected some more data, which can be helpful:



Also, Merry Christmas to everybody in this Forum !

[Iacopo]

The wobble appearing in spinning tops after the resonance is not precession nor simple nutation.
It is something more complicated.  Probably something related to nutation, like in spindulums.
I have no more ideas.  I am going to write the conclusions of this thread.

[Iacopo]

CONCLUSIONS

This was an interesting thread for me. 
I am sorry that my exposition was a bit chaotical here and there; the experiments have led to some discoveries, but also to various new doubts and questions.
The matter turned out to be more complicated and difficult than I thought at first.
I resume here what I have learned.


RESONANT FREQUENCY

Spinning tops and spindulums have a natural resonance frequency, a frequency to which they are willing to wobble by resonance.
The nature of this natural frequency is not simple.

IN SPINDULUMS it is related to both circular oscillation and nutation.
These two movements are strictly linked together, and, if the top spins and the tip is not at the center of mass, the two movements always compare together, and it is impossible to observe them separately. The oscillation and the nutation have always the same direction, which is the direction of the spin. The effect of the oscillation is that to make the nutation faster, (or, the nutation makes the oscillation faster, if you prefer so).  At very low spin speed the oscillation is the dominant component, at medium and high speed instead nutation is the dominant component, especially when the tip is near to the center of mass. If the tip is exactly at the center of mass, the nutation is pure, and the oscillation absent.
The natural resonance frequency is not fixed, it changes continuously with speed, becoming lower at lower spin speed.

IN SPINNING TOPS the natural resonant frequency seems related to nutation, like in spindulums.
It is not simple nutation anyway, there is something altering its speed, like in spindulums.
In spindulums, THE NATURAL RESONANT FREQUENCY IS ALWAYS HIGHER THAN THE PURE NUTATION FREQUENCY.
In spinning tops instead, THE NATURAL RESONANT FREQUENCY IS ALWAYS LOWER THAN THE PURE NUTATION FREQUENCY.
My personal idea about it is that the cause of the altered speed of the nutation could be the same in both cases:
the oscillation makes the nutation faster in spindulums.
In spinning tops there cannot be oscillation but maybe there is its negative counterpart, a force related to gravity, trying to tilt them outwards, (instead of inwards like in spindulums).
The reversed direction could maybe explain why the alteration makes the nutation slower, and not faster, in spinning tops.   

 
DRIVING FREQUENCY

The driving frequency is given by the top spinning while being unbalanced.
The unbalance makes the top to wobble with the same rotational frequency of the spin speed;
the spin speed decreases by the time; when the spin rotational frequency is going to approach the resonance frequency of the top, the top starts to wobble more and more intensely.
This wobble, (unbalance), is producing the second wobble, (nutation/oscillation), by resonance.
 
In the beginning of the resonance phase, the two wobbles have the same speed, so we can't distinguish them;
but, soon, the two speeds start to differentiate, and the wobble of the top becomes intermittent and less intense.
The intermittent wobble tells the coexistence of two wobbles of a different nature, having different frequencies.
The second wobble fades away, in a shorter or longer period of time.


PHASE SHIFT

While spinning unbalanced, the stem of the top wobbles, with the same rotational speed of the top, (or spindulum), staying tilted towards the light side or towards the heavy side of the top. 

What side the stem stays tilted towards depends on the relative speeds of the two wobbles:

if the driving frequency, (unbalance), is lower than the resonant frequency, the stem stays tilted towards the heavy side of the top, (phase 0°).

But if the driving frequency is faster than the resonant frequency, the resonant wobble can't keep pace with the driving wobble, and a delay up to 180° develops between the two wobbles, (phase 180°);  In this case, the stem stays tilted towards the light side of the top.

During the resonance there is a phase shift, because the faster wobble changes, (graphics below).

In unbalanced spinning tops the stem stays tilted towards the heavy side at frequencies above resonance, and towards the light side at frequencies below resonance.
In spindulums the opposite happens, the stem stays tilted towards the light side at frequencies above resonance, and towards the heavy side at frequencies below resonance.

 

 

[Iacopo]

Do you like headaches ?
I discovered a new weird and bizarre behaviour of spindulums, which I am going to show.
I have no idea at all why they behave so.
First I have to make a short video about it, maybe today or tomorrow.

[ortwin]

Do you like headaches ?
....

This type of headaches I like most, when the pain subsides because we found an explanation that we can understand and that we agree on.
I am curious what it will be. Given the title of this thread, a certain type of phenomenon can be expected. Hopefully something I can also observe with my equipment.

[Iacopo]

Hopefully something I can also observe with my equipment.

It will be not difficult to see it, if you have a spindulum..

[Iacopo]

For this test I used the spindulum "B", but similar results are obtained also with the other spindulums.



We have seen that a spindulum can oscillate in circular motion, without spinning.
It is possible to make the spindulum to oscillate and spin at the same time, in which case the spin makes the circular oscillation faster;
this is when the spin and the circular oscillation have the same direction.

But what if I spin the spindulum in the opposite direction of the circular oscillation ?
I tried, it works, and I discovered the following facts:

In opposite direction, the spin makes the circular oscillation slower and slower, at the increasing of the spin speed.

When the spindulum spins very slowly or it doesn't spin, the nature of the oscillation, (like in a pendulum), is more evident.
But when the spin speed increases, it becomes obvious that, what at slow speed appears like circular oscillation, at higher speed is nutation when spin and oscillation have the same direction, and precession when they have opposite direction instead. Isn't that weird ?

I plotted the data in the graphic below.
Practically, the curve of nutation and the curve of precession have the same origin, which corresponds to the natural oscillation frequency of the spindulum.



In the video you can see how the spindulum moves, for more clarity. 
P2 and P6 are precession, N4, N7 and N9 are nutation; the speed data extrapolated from this and other videos of this spindulum were used for the graphic above.  The spindulum is not unbalanced, (the added plasticine at one side of the spindulum is for to make it balanced), so there is not unbalance wobble here, which would have made the things a bit more confused.

https://youtu.be/OG_YqnaIfrU

[ta0]

Very nice experimental work, Iacopo. 

I reproduced the qualitative results using my Maxwell Dynamical Top (that can easily be converted to a Spindulum by adjusting the tip screw). I haven't yet checked the "resonance", i.e. that the pendulum frequency is the limit that separates the precession and nutation frequencies.

At slow speed I saw what you say: when the spin and rotation are in opposite directions it can precess steady but if you try to make them go in opposite directions it will nutate strongly (I think strictly speaking it's a nutation superposed to a precession, but they cannot be differentiated as they are of similar amplitude and frequency).

I think that when the spindulum is spinning fast, the regular equations of a fast top apply. The behavior is what you expect from conservation of angular momentum. If the top is straightening up, it has to precess backwards to cancel the increased vertical angular momentum due to the spindulum aligning it's spin with the vertical direction.

[ta0]

A fast measurement:

Pendulum period in two perpendicular axes: 7.8 sec
Circular pendulum period: 7.9 sec

Circular period with a small counter spin: 6.5 sec
Circular period with a small forward spin: 9.8 sec (less precise measurement)