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[Russpin]

The period of a torsional pendulum is given by:

T = 2pi*sqrt(I/K)

where I is the moment of inertia and K is the torque constant.

For your torsional pendulum, K is constant, So the period is proportional to the square root of the moment of inertia.
So your pendulum period changes with the length of the cylinder because of the mass and thus the moment of inertia changes.


For a trifilar pendulum, K is proportional to the weight applied to it so K = a*m*g where a is a constant, m is the mass of the object and g is the acceleration of gravity. So

T = 2pi*sqrt(I/(a*m*g))

but I is proportional to m, letting I = mr² where r is the radius of gyration. So

T = 2pi*sqrt(m*r²/(a*m*g)) = 2pi*r*sqrt(1/(a*g))  note: mass cancels out

So the period of a trifilar pendulum is proportional to the radius of gyration, not moment of inertia.

The radius of gyration of a cylinder does not change with mass (or length) so

The period of a trifilar pendulum does not change with length or mass of the cylinder.

[ta0]

Exactly what my point was.

[Russpin]

So do you still think the period of a trifilar pendulum changes with the height of the cylinder ?

Edit: Sorry ta0 I guess I misread you post. I thought you said that the period of the trifilar pendulum should change with cylinder height.

[Iacopo]

My calculated I₁ is still a little low at CM,  I1 = 3.4x10^-5 Kg m².

The difference is about 6 %. 
This is not strange, because I don't have an accurate way for measuring the height of the center of mass on the tip, also the toppling down speed is a bit difficult to measure accurately because the top wobbles a lot in the end of the spin.  But the formulas are interesting. 

[Iacopo]

That's a surprisingly (to me) precise result on the calculation of the critical speed! 

.. the moment of inertia. On the torsional pendulum you get it directly

Now I am curious to try the formula with other tops...

How is it made the torsional pendulum, to be sensitive to weight ?  It is the top suspended to a simple string ?

[ta0]

I thought you said that the period of the trifilar pendulum should change with cylinder height.

Ha, just the contrary. I was pointing the difference between the trifilar pendulum, where a change in moment of inertia due to a change of the cylinder height does not affect the oscillation period, and the torsional balance, where any change in moment of inertia will change it.

How is it made the torsional pendulum, to be sensitive to weight ?  It is the top suspended to a simple string ?

I guess you meant insensitive to weight. Yes, a simple string and the torsion is given by the mechanical properties (shear modulus) of the string material (and length/diameter). As long as you do not twist it too much (i.e. below the elastic limit), the torsion torque will be proportional to the twist. And if you do not put too much weight the torsion constant will remain constant.
My original post is here: Measuring the Moment of Inertia

[Iacopo]

I guess you meant insensitive to weight. Yes, a simple string and the torsion is given by the mechanical properties (shear modulus) of the string material (and length/diameter).

I thought to weight in the sense that weight is part of the moment of inertia, so if the torsion pendulum measures directly the moment of inertia, in this sense, it should be sensible to weight.
I was a bit confused at first when you writed that it measures the moment of inertia (and not the radius of gyration), but now I see it makes sense instead.

[Jeremy McCreary]

Calculating moments of inertia from topple speed measurements is very risky business -- not just in the cases discussed in this thread, but in general.

Maybe this top could spin for some more time and for some reason it topples down sooner ?
There is a bit of imbalance wobbling at the end of the spin, maybe this makes the top to topple sooner, but I am not sure.
H is a bit lower than I thought. I checked it again, more accurately, it is between 5.8 and 6.2 mm.

That spread in H (tip-CM distance) alone introduces significant uncertainty in any moment calculation. Happy to show the math if anyone's interested.

Like you, I'm often left to guess CM locations in the tops I make, and that alone is reason enough to distrust moments of inertia from topple speed measurements. But the practical problems don't end there. Measuring topple speed can also be tricky business.

First, to amplify an earlier point, critical speed formulas don't predict topple speed. They merely give its upper bound -- i.e., the highest speed at which a top can fall over in response to a small perturbation (e.g., a breeze or a tip irregularity). When the top actually falls over, at or below critical speed, depends in large part on potentially imperceptible environmental fluctuations over which one may have little control.

What then do you actually see when a steadily and smoothly precessing or sleeping top finally spins down to critical speed? Quite possibly nothing at first, but eventually, high-amplitude nutation (oscillation in tilt angle) will appear and then progress against a backdrop of decaying spin rate and growing precession rate and angle. After that, it's just a matter of time until the growing total tilt consumes all available ground clearance and puts the rotor on the ground.

So, do we take "topple speed" at the time of the easy-to-recognize rotor strike, at the harder-to-recognize onset of high-amplitude nutation, or somewhere in between? It may not matter much when these events play out in a fraction of a second at low spin decay rate, as they certainly do in some real tops. But some of my tops take well over a second to hit the ground after they begin to show what I consider to be "high-amplitude" nutation, and their spin decay rates are generally much greater than the ones Iacopo's tops tend to enjoy. What then?

But suppose you come up with a measurement method you can live with. Statistically, you'd then have to measure many topple speeds under well-controlled initial and environmental conditions just to estimate their upper bound, critical speed. I have yet to come across a data set like that, and no wonder given the bother involved.

Finally, these practical issues arise under any definition of "speed" about the symmetry axis -- whether total angular speed (w₃) or pure spin rate (s). When the (precession rate / spin rate) ratio is much less than 1, as it often is at the time of release, the definition of speed may not matter. But this important ratio is necessarily at its greatest during the toppling process. Plugging a measured toppling spin rate s into the critical w₃ formula might then make a significant difference in the calculated moment of inertia.

[Jeremy McCreary]

Calculating moments of inertia from topple speed measurements is very risky business -- not just in the cases discussed in this thread, but in general.

Should've mentioned that when estimating I₁ from topple speed, an X% error in the value used for I₃ has just as much impact on accuracy as an X% error in the value used for topple speed. The impact of an X% error in the value used for H is less but can still be significant.

[Russpin]

Calculating moments of inertia from topple speed measurements is very risky business -- not just in the cases discussed in this thread, but in general.

I like to live dangerously !

This was just an experiment. It was never meant to be a serious method for calculating I₁. However I have a theory that Iapopo's tops will follow the critical speed formula much more closely than regular tops. But as you point out, this may be too difficult to verify empirically. I really should  test my theory with numerical simulations first before even attempting an empirical validation. 

[Iacopo]

I ended the wear test, the result has been a bit surprising for me.

Generally my tops with a spiked carbide tip, after about 5 - 10 hours of spinning, start to wobble (nutation) spontaneously, because of weared contact points.
Here an example, the top at the right is wobbling because of weared contact points:

https://youtu.be/kXDKc8vOz3M?t=102

This is why I started using often HHS spiked tips, which are quite less prone to this problem.
The wobbling top in the video weighs 160 grams.
My two latest tops weigh both 119 grams.  I wanted to see if the reduced weight would help to decrease wear of the contact points using the carbide tip.

The result is that after 50 hours of spinning the top was still spinning smoothly, without any wobbling, still perfectly vertical at medium and high speed. This is better than I hoped.

I observed the base (made of tungsten carbide) at the microscope and saw this:



This is a mm 0.06 hole (deep about mm 0.02); it is a littler hole than that of the top in the video, but I started wondering if the reduced weight of the top could be the only cause of the absence of any spontaneous nutation wobbling.

I cleaned the base from the oil and tried to spin the top without oil: this usually facilitates the spontaneous nutation in a top spinning in a holed base. 
But still no wobbling at all. The top spinned very well with no problems.

I tried to force a nutation kicking the stem of the top with a finger:
the top started nutating but the nutation disappeared rapidly, it lasted less than half a minute, then the top spinned smoothly again, without any wobbling.

When I do this with a top with deeply recessed tip, the nutation lasts for longer time, (two, three minutes or even more), even if the contact points are in perfect conditions. 

I have come to think that the cause is probably related to the height of the center of mass:
if CM is at the contact point, or very near to it, nutation in the flywheel happens in the same way as in a flying nutating object, there isn't any strong opposition to nutation.

But if the tip is external, at some distance from the CM, then the tip has to slip on the spinning surface, for the top to nutate about the CM.  Or, the tip doesn't slip, but in this case the CM has to move in circular motion, during a nutation, but the CM tends to be steady, (inertia), so I think that in this situation there is a natural resistance against nutation, and this is why this top refuses to wobble spontaneously.

I made efforts in the past to solve this problem, looking for harder and more wear resistant materials for the contact points. I made various tests, spent time and money.. without finding a definitive solution.

Now a solution has come, accidentally...   

[Russpin]

Guess we'll have to agree to disagree on the best use of "spin". The (many) sources I've read generally don't use "spin" for w₃. Instead, they reserve it for your "psi dot", which is my s (lower case).

All my sources use "spin" to mean w₃

The the equations of motion from this paper are the ones I used to make the spinning egg and 5 degree of freedom top simulations.

http://www.damtp.cam.ac.uk/user/hkm2/PDFs/Moffatt_Shimomura_ea_2004_PRSA_460_3643.pdf

Look at the text under equation 2.1

It clearly defines spin as w₃ .

[Jeremy McCreary]

Guess we'll have to agree to disagree on the best use of "spin". The (many) sources I've read generally don't use "spin" for w₃. Instead, they reserve it for your "psi dot", which is my s (lower case).

Look at the text under equation 2.1
It clearly defines spin as w₃ .

Yes, the authors of this excellent article use "spin" in the way you prefer.

Took this opportunity to reread the whole thing. I can generally follow at least the gist, but I don't understand their concept of "gyroscopic balance" as defined in Eq. 4.1. Nor do I understand how they got there from Eq. 3.10. Note that I'm not questioning the result, just wondering about the physical meaning.

The authors describe gyroscopic balance as a "high-spin state" (accepting for the moment their use of "spin"), but it's hard to imagine a real spinning spheroid of reasonable CM height and moment ratio (in their nomenclature, A / C) meeting the condition (Eq. 4.1) without an unusually high ratio of precession rate (phi_dot = capital omega) to psi_dot. What am I missing here?

[Russpin]

Nor do I understand how they got there from Eq. 3.10.

Equation 4.1 is derived from the second equation of 3.10 (2). The right side of 3.10 (2) is set to zero because the frictional and gravitational effects are ignored. The inclination changes slowly with time, so the lambda term on the left is zero. Dividing out the omega sin theta gives equation 4.1.

The authors describe gyroscopic balance as a "high-spin state" (accepting for the moment their use of "spin"), but it's hard to imagine a real spinning spheroid of reasonable CM height and moment ratio (in their nomenclature, A / C) meeting the condition (Eq. 4.1) without an unusually high ratio of precession rate (phi_dot = capital omega) to psi_dot. What am I missing here?

They are using the term “spin” very generally here. At time zero theta is 90 degrees so n is zero. As the spheroid rises, theta goes to zero and n increases while omega decreases.