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[Jeremy McCreary]

It's often said around here that hollow throwing tops outperform solid ones. Yet in theory, hollowing might help in some ways and hurt in others if the tip and external size, shape, and surface roughness all stay the same.

So, curious as to exactly how the trade-offs involved play out in real use? In exactly what ways do hollow tops excel over solid ones in practice? Are there practical situations where solid tops work equally well or even better?

[ta0]

I can think of many situations where a hollow top outperforms a solid one. The obvious one are all climbing tricks, where any extra weight is nefarious.
Tricks in which the top is reversing the direction of travel (e.g., short circuit) are made more difficult by the inertia of the top (you have to pull back stronger when boomeranging a solid top). I guess even a trick in which the top does a circular motion could be made more difficult because the extra centrifugal force (although I don't remember trying corkscrew with a solid top).

The only obvious case that I can think in which a solid top is beneficial is in battling tops 

But I'm intrigued by long wire walkers. Does the linear inertia of a solid top help? My guess is no, but I'm not sure. The Chinese use solid tops. I cannot recall if Cecil used some hollow tops for his longer ones.

[Jeremy McCreary]

I can think of many situations where a hollow top outperforms a solid one. The obvious one are all climbing tricks, where any extra weight is nefarious.
Tricks in which the top is reversing the direction of travel (e.g., short circuit) are made more difficult by the inertia of the top (you have to pull back stronger when boomeranging a solid top). I guess even a trick in which the top does a circular motion could be made more difficult because the extra centrifugal force (although I don't remember trying corkscrew with a solid top).

The only obvious case that I can think in which a solid top is beneficial is in battling tops 

But I'm intrigued by long wire walkers. Does the linear inertia of a solid top help? My guess is no, but I'm not sure. The Chinese use solid tops. I cannot recall if Cecil used some hollow tops for his longer ones.

Interesting! These seem to be cases where the main issue is too much or too little mass or linear (translational) momentum. Absolute AMI (as opposed to AMI per unit mass) may also enter the picture in the battle top case.

[cecil]

I making them - I know.  Hallow tops are faster and they perform much better. You can't do to much with a 10.0 inch solid top.

[Jeremy McCreary]

I making them - I know.  Hallow tops are faster and they perform much better. You can't do to much with a 10.0 inch solid top.

Good point. So playability demands hollowing above some critical diameter that depends on the material and the player.

[Jeremy McCreary]

Nerd Alert! A little math exercise to get a feel for how hollowing affects mass properties...

Take a simple "blank" -- a perfect solid right circular cylinder of radius R, axial length L, and uniform density b -- and hollow out its center to inside radius R_i. Before hollowing, R_i = 0. After hollowing, we're left with a "tube" with R_i < R.

The tube's mass M, axial moment of inertia (AMI, I₃), and specific AMI (AMI per unit mass, J₃) can be expressed in terms of the corresponding blank properties and the "radius ratio" X = R_i / R, like so...

M = M₀ (1 - X²)
I₃ = I₃₀ (1 - X⁴)
J₃ = I₃ / M = J₃₀ (1 + X²)

where the subscript "0" refers to the blank, and the subscript "3" refers to the symmetry axis.

So, hollowing the blank reduces its mass by a factor of 1 - X² and its AMI by a factor of 1 - X⁴ while increasing its specific AMI by a factor of 1 + X². High specific AMI is generally a very good thing in tops of all kinds.

The factors above don't depend on the blank's axial length. For the tall black and yellow tubular rotor below, M = 114 g, R = 65 mm, L = 56 mm, R_i = 57 mm, and X = 88%. Relative to a solid rotor of the same outer radius, axial length, and density, this hollow rotor has 23% of the mass, 40% of the AMI, and 177% of the specific AMI.



You can really feel the high AMI when you twirl this thing by hand. Spin times by hand and with a suitable starter are ~45 and ~70 sec, respectively.

I can think of many situations where a hollow top outperforms a solid one. The obvious one are all climbing tricks, where any extra weight is nefarious.
Tricks in which the top is reversing the direction of travel (e.g., short circuit) are made more difficult by the inertia of the top (you have to pull back stronger when boomeranging a solid top). I guess even a trick in which the top does a circular motion could be made more difficult because the extra centrifugal force (although I don't remember trying corkscrew with a solid top).

These all sound like reasons to lean toward lower mass and higher specific AMI in non-battle tops. As an added bonus, lower mass also reduces tip friction.

[ta0]

Thanks Jeremy.

I have to put a caveat to what I said before. Although in practice it doesn't seem to happen, in theory if you hollow it too much (walls too thin) you could end up with at top that is too light to play. For example, in many regeneration tricks a heavier top facilitates the string feeding with the non-throwing hand, which is an essential part of the trick. Also, a very small moment of inertia would limit the energy storage at reasonable spin rates (thus increasing the relative importance of air drag).

Another interesting comparison, is between two tops, one hollow and one solid, but with the same mass (and therefore different materials). Here there is absolutely no doubt that the hollow one is superior. Not only its specific moment of inertia is close to 2 times the solid one, as in your derivation for the hollowed out top, but also its absolute moment of inertia is double.

[Jeremy McCreary]

I have to put a caveat to what I said before.

Thanks, I'm really trying to understand this hollowing business from a practical point of view.

Although in practice it doesn't seem to happen, in theory if you hollow it too much (walls too thin) you could end up with at top that is too light to play. For example, in many regeneration tricks a heavier top facilitates the string feeding with the non-throwing hand, which is an essential part of the trick.

Makes sense that there would be both a lower and upper limit to mass WRT playability -- at least for some tricks. From the recent Strommel8 Love thread, I gather that the Love's mass -- presumably smaller than average -- might be pushing the lower limit for some players and some tricks.

Then there's the loss of durability with overly thinned walls.

Also, a very small moment of inertia would limit the energy storage at reasonable spin rates (thus increasing the relative importance of air drag).

Makes sense. That's another reason to favor high specific AMI -- to get the most AMI you can from the mass available.

One of the things that makes hollowing interesting to me as a rank beginner is that string-top and hand-top interactions greatly complicate the observed behavior and playability even when the mass properties are well understood. For example, there must be some interesting trade-offs involving mass and absolute AMI when it comes to the spin rate at the moment the top leaves the string. Finger top dynamics are a lot simpler.

Another interesting comparison, is between two tops, one hollow and one solid, but with the same mass (and therefore different materials). Here there is absolutely no doubt that the hollow one is superior. Not only its specific moment of inertia is close to 2 times the solid one, as in your derivation for the hollowed out top, but also its absolute moment of inertia is double.

Very helpful.

Specific AMI (AMI per unit mass) doesn't come up much around here, but I think it deserves more attention. It's just the square of the axial radius of gyration (ARG), which we have discussed from time to time, but I find ARG much less helpful than specific AMI when it comes to top design and play. For example, it's easier for me to think -- qualitatively of course -- about how much AMI I'd like to get for a specific amount of mass (or material) and design accordingly.

[ta0]

From the recent Strommel8 Love thread, I gather that the Love's mass -- presumably smaller than average -- might be pushing the lower limit for some players and some tricks.

A Love weighs 47 gr, which is slightly more than a King Cobra (with metal rim) which weighs 45 gr.
The problem of the Love, in my opinion, is the relative low moment of inertia. Not only it has a smaller diameter (44mm against 54mm) but a lot of the mass is around the tip section.

[Jeremy McCreary]

A Love weighs 47 gr, which is slightly more than a King Cobra (with metal rim) which weighs 45 gr.
The problem of the Love, in my opinion, is the relative low moment of inertia. Not only it has a smaller diameter (44mm against 54mm) but a lot of the mass is around the tip section.

Ah, so the problem with the Love is not so much low mass as it is low specific AMI.

[Jeremy McCreary]

Edit: Rewrote some of this to make it clearer and to emphasize practical applications, but the update got lost in the recent server crash.

Nerd Alert! To finish my little math exercise, I looked at the effect of hollowing on the tranverse moment of inertia (TMI) and specific TMI. Unfortunately, this part of the story is more complicated, and for that reason, I'll limit the discussion here to TMI about the CM rather than the tip.

Doing the math: We start once again with a solid cylindrical blank of radius R, axial length L, and uniform density b and hollow out its center to form a tube with inside radius R_i. In the first installment, we found that the tube's mass, AMI, and specific AMI could be expressed rather simply in terms of the corresponding blank properties and the "radius ratio" X = R_i / R, like so...

M = M₀ (1 - X²)
I₃ = I₃₀ (1 - X⁴)
J₃ = I₃ / M = J₃₀ (1 + X²)

where the subscript "0" refers to the blank, and the subscript "3" refers to the symmetry axis. It's worth noting that 1 - X⁴ is always greater than 1 - X² in hollow tubes, where by definition 0 < X < 1.

To apply the same approach to the tube's TMI about the CM, we need another key proportion -- the "aspect ratio" A = L / R. Then

I_1cm = I_1cm0 [3 (1 - X⁴) + A² (1 - X²)] / (3 + A²)

where the subscript "1cm" refers to any axis passing through the tube's CM perpendicular to the symmetry axis.

For pancake-like low-aspect tubes with A << sqrt(3) = 1.732, as in the big yellow rotor below (X = 0.82, A = 0.19), the way hollowing affects TMI is largely independent of aspect ratio. In these cases,

I_1cm ~ I_1cm0 (1 - X⁴),

which varies with X just as AMI does after hollowing.



But as A increases to "taller" values more typical of throwing tops (A ~ 2), the TMI about the CM shifts progressively toward

I_1cm ~ I_1cm0 (1 - X²),

which varies with X as mass does after hollowing.

For specific TMI about the CM,

J_1cm = I_1cm / M = J_1cm0 [3 (1 + X²) + A²] / (3 + A²),

which increases with radius ratio X for any aspect ratio A. For low-aspect tubes with A << 1.732, this reduces to

J_1cm ~ J_1cm0 (1 + X²),

which varies with X just as specific AMI does after hollowing. Again, the way hollowing affects TMI is largely independent of aspect ratio. But taller aspect ratios progressively slow the rise of J_1cm with X.

Some physical implications: For starters, hollowing has roughly the same effect on TMI about the CM as it does on AMI in pancake-like low-aspect tubes. Ditto for specific TMI and specific AMI. In such cases, the TMI/AMI ratio about the CM will change very little with hollowing. But at throwing top-like aspect ratios on the order of A ~ 2, hollowing tends to reduce TMI/AMI. This important ratio has a strong but complicated impact on top behavior, including responsiveness. Perhaps Jorge will have something to say about TMI/AMI in throwing tops.

In contrast, the way hollowing affects mass, AMI, and specific AMI depends only on radius ratio. Aspect ratio has nothing to do with it in blanks with cylinder-like symmetry, regardless of the lateral profile.

CM location: Finally, what does hollowing do to CM location? Well, boring out a cylindrical blank can't shift the CM from its original location. In non-cylindrical blanks, the CM will shift toward the blank's widest diameter. And if we add a tip and a stem and some spokes to turn the tube into a finger top, the top's overall CM will lie somewhere between the collective CM of all the additions and the CM of the tube itself.

[the Earl of Whirl]

That's exactly what I was thinking!

Actually, I wasn't thinking about any of this before but it sure sounds impressive and looks quite fabulous with all the formulas and numbers.  I will leave it for someone to have a better response but for right now I will simply say......

Good work!!!

[Jeremy McCreary]

That's exactly what I was thinking!

Actually, I wasn't thinking about any of this before but it sure sounds impressive and looks quite fabulous with all the formulas and numbers.  I will leave it for someone to have a better response but for right now I will simply say......

Good work!!!

Thanks, MIke! Meant this to give our nerdier top-builders some insight into what hollowing actually accomplishes. Unfortunately, I'm not good at explaining such things without math.