Measuring the Moment of Inertia

[ta0]

For a long time I had been wanting to assemble a setup to measure the rotational inertia (also called Moment of Inertia) of our tops. It must be one of the most fundamental specs of a spintop but I never see it mentioned on spintop reviews (even though any CAD software used to design a top should spit it out). Is it on yo-yos?

The Caltech lab instructions that I found while searching for the Maxwell Top describes a way of doing it using a torsional balance, so I decided to give it a try and see how difficult it was. To my surprise, it worked better than I expected so I went ahead and measured a few tops.

I replaced the torsion rod of the lab notes for a piece of fishing line (450 mm of 20 lb Test mono-filament). All you need is the line, a place to attach it and a stopwatch. Nothing more. This is my setup:



The measurement is very simple: you rotate the top in one direction, so the line twists, and let go. The square of the time it takes for the top to twist back and forth is proportional to the rotational inertia.

If you want to take absolute measurements you just need to calibrate it using an object with a known rotational inertia, such as a solid rod (what I did).

I measured Spintastics tops because the hole on the crown made attaching the fish line easy. These were the results:



Spintop Moment Inertia


QS full metal6.20


QS with TG cap4.31


HP with bulldog cap3.61


TG with ring3.26


TG no ring2.99


QS cap2.49


Bulldog cap0.53



The units are kg m2 x 10-5.

[Pulpowsky]

Any explanations from the engineer? What the moment of inertia do for us? etc, etc.

[Spinningray]

The higher the moment of inertia, the more force it takes to speed up the spin. It also takes more force to slow it down, so it should spin longer.

[ta0]

When you add rim weight to a top (or a yo-yo) what are doing is increasing its Moment of Inertia (with respect to rotation axis).

I just realized there is an easy way to attach any of our tops to the torsional balance. Because the tip is at the axis of rotation it does not add much to the rotational inertia: if the tip is removed it won't change much. So we can use a plug to attach the fishing line at the tip end of the top and hang it upside down. Tops with open or no crowns will be as easy to measure as regular tops.

Technical notes:

The torsion balance works similar to a pendulum. The time it takes to "swing" (twist) back and forth does not change with the amount of twist ("amplitude") because if you twist it a larger angle it just swings faster and the "period" remains constant. This is true as long as you don't over-twist the line. I started the balance with a twist between 1 and 2 full turns. On my first measurements I used a camera to record the swings and check that the period did not change as it lost energy and the swings got smaller. It did remained constant. After I got convinced of this I just used a stopwatch and timed 10 changes in direction.

For calibration I used rods, both, hanging vertical (low moment of inertia) and hanging horizontal attached to their center (high moment of inertia). I covered half-periods (i.e., time between direction switches) of 2.3 seconds to 66.7 seconds that correspond to a factor of over 800 times in moment of inertia. The spread of calibration constants from 7 rods was just +/- 2.5% what I found incredible.

[poptop]

How about your Gasing...

Nice work Ta0.

Are the moments additive? Does the inertia of the QS cap plus the QS body equal the assembled parts?

Are there any other factors that influence moment of inertia besides 1. total mass and 2. the distance/distribution of this mass away from the axis of rotation?

[the Earl of Whirl]

I feel like I am making an expression just like my dog when she doesn't want to come in the house. I yell at her and yell at her and she just gives me a blank stare while smiling politely.

Ta0, this is great information. I am giving you a blank stare while smiling politely at you!

[ta0]

How about your Gasing...

The line is 20lb Test, so it should be able to hold the gasing if I find a good way of attaching it. But it may stretch the line enough to lose the calibration, so I would need to recalibrate with a heavier cylinder. Perhaps one day . . .

Are the moments additive? Does the inertia of the QS cap plus the QS body equal the assembled parts?

Yes, they should add. That is a good idea to verify the data.

Are there any other factors that influence moment of inertia besides 1. total mass and 2. the distance/distribution of this mass away from the axis of rotation?

No, it is just the mass times the square of the distance to the axis of rotation (calculated for every point of the object and added together).

Mike, are you saying you are getting old like your dog?

[RotatED]

Fascinating!
As for: "more force it takes to speed up the spin"... please explain.
If I have two identical tops, one with 2x the moment of inertia, what does it mean in terms of throwing each and achieving the same RPM?
Must one be thrown harder, or with a longer string, etc.?
Thanks.

[poptop]

If you've ever tried to spin a car or motorcycle tire while it's off the ground, you'd notice that it takes more force to get it going compared to a bike tire; likewise, once it gets going, its harder to stop.

I think this is true for our tops of various moments of inertia--some take more force to get going. Adding weight rings increases the moment of inertia.

It would seem that there might be some ideal ratios of moi to total mass for fixed vs. beating play?

As you get higher and higher moi, it will be harder and harder for string friction to add spin right?

[ta0]

Poptop's example is a very good one.
Regarding tops, you need to apply more torque to a snap-start with a spintop of higher rotational inertia. But once it is on the hand it will last spinning longer.
How it affects play on the string is a more complicated question as this also depends on the height of the center of mass and the total mass, both of which change if you add a weight ring.

As you get higher and higher moi, it will be harder and harder for string friction to add spin right?

I would say so, if you keep the same tip size. More massive tops generally require a bigger tip so the string can provide more torque.

[pplgrande]

This is some interesting stuff, thanks for explaining
Honestly I have read this topic over and over and while some I undestand, most of it is confusing
Im going to keep reading it until get it, this is very interesting

So whats ideal? smaller, lighter, bigger, heavier?

[ta0]

So whats ideal? smaller, lighter, bigger, heavier?

In general you want a higher rotational inertia, or at least a higher ratio to weight. That is why hollow tops and tops with weight rings tend to be better than solid tops.

Are the moments additive? Does the inertia of the QS cap plus the QS body equal the assembled parts?

Yes, they should add. That is a good idea to verify the data.

Well, I measured the internal weight ring of Trompo Grande: it has a radius of 26 mm and and weight of 9.5 gr. Its moment of inertia is just M x R2 = 0.64 in the same units as before. On the table if you subtract TG with ring minus TG without ring you get 0.27, or a little less than half that. I guess there is some measurement error there (error / TG with ring = 11% relative error). Next weekend I'll try to measure it more precisely.

[sixtoe]

ta0. I don't want to seem arrogant or condescending but in the interest of science I need to point out a flaw in your measuring methodology regarding moment of inertia measurements in regards to defining play characteristics of spintops.

One missing measurement is the center of gravity. This along with a measurement of a 2nd moment of inertia will give 3 data points that could be correlated among other sets to make an interesting chart that would show definitively the similarities and differences in spintops. It chart could even be correlated against player preference to model a 'perfect' spintop's data set.

Currently the experiment has only measured the MOI of the rotational axis. The MOI measurement of the rotational axis by itself only affects resistance to initial energy input, maximum RPM in regards to maximum energy input, and resistance to losing inertia through friction. While this is a very important measurement it is not the one that I think affects play the most.

In my opinion the MOI that affects play the most is perpendicular to the rotational axis. This measurement gives is needed to calculate the resistance to precession where the measurement of the moment of inertia along the rotational axis alone will not. I must point out that for the measurements of this MOI across subjects must be made using the CG as the suspension point of your torsional MOI measurement instrument. If you use a common center point along the length of each subject the measurements will not be corrected for the affect of CG. It is noted that by compensating for CG in the measurement will also give the measurement of CG by default. You will need to build a suspension frame that will allow measurement of CG and a suspension point that adjusts to the CG.

Hypothesis: Having all 3 measurements, 2 of MOI along perpendicular ax?s, and 1 of CG will give a complete set that will model a top's play characteristics and could be used to correlate among multiple subjects in a nifty chart. It will also allow a plot of resistance to precession that follows RPM decline vs maximum spin time starting with a top RPM.

If I have made any error in logic or reasoning here I insist they be pointed out by anyone who understands what I have presented here so that I may adjust my hypothesis.

Thanks for reading,
Sixtoe

[ta0]

Hey sixtoe, nice to see you are back. And this post may encourage me to make more moment of inertia measurements, as I had promised to do but put in the back burner.

The position of the center of mass is something we have discussed from a long time. I posted a simple way of locating it on this thread and pizquie (Pedro Izquierdo) from Spain made an apparatus specifically for that purpose.

Yes, the moment of inertia on an axis perpendicular to the axis of rotation is also an important factor to how the top plays as with the other two determines the rate of precession, as you said. On the throw it should be with respect to an axis through the center of mass and during play with respect to an axis through the tip. Of course, knowing one the other can be calculated using their distance.

I think the perpendicular MOI is the logical next step in these measurements and I was planning to do it once I got around finishing the axial MOI measurements. The method would basically be the same but more burdensome as the line needs to attach at the height of the center of mass.

I would love for somebody else to help me do this with the most popular tops. Can you do it?

[ta0]

I need to make a correction to the above. The rate of precession (in steady state) does not depend on the MOI perpendicular to the spin axis! I checked on Crabree's book and the rate of precession for a top spinning with the axis horizontal is equal to: p = T/(I.w), where T is the torque from gravity, I the MOI with respect to the spin axis and w the spin rate

I guess the explanation is that if the precession is constant there is no relevant "acceleration". I believe that during play the perpendicular MOI will affect play as the top is continuously changing movement, but it is difficult to say how much.