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[Jeremy McCreary]

It's amazing that these offset square tops made with the constraints to discrete steps of LEGO, can spin so smoothly.

Agree, amazing that LEGO-friendly solutions exist. Used a spreadsheet based on the formulas above to find doable combos of small rectangle dimensions and areal densities, with or without the CM on the boundary. The 3 tops I've shown here pretty much cover the possible square solutions using studded LEGO parts.

There is something I don't understand. The golden ratio black top needs a density radio of 3.6. But on the photos it looks like the heavy side has three layers, with the bottom layer carved out, so I would expect it to be more like 2.6 

Black isn't a golden square offset top -- just the closest I could get. Its density ratio is 0.360, not 3.6. That means that the small black rectangle is 1 / 0.360 = 2.78 times denser than the larger one.

The algebra behind the formulas was much nastier than expected. Defining the density ratio as I did simplified the math a bit, but using its reciprocal instead would have caused a lot less confusion.

[ortwin]

Looking forward to see a liitle video on these!

...
Note on White's tip: Switching White's red tip holder from the taller to shorter version below lowered the top's CM by 6 mm. This alone bumped spin time from 50 to 70 s (when it remembered to fall).
...
Tip design is one of the harder parts of LEGO topmaking. Any wiggle in the tip assembly will show up as wobble in the top, and suitable contact surfaces are few and far between.

 
 
When doing some cleaning today in my son's room, I accidentally came across the blue  LEGO part you can see in this picture: 

I instantly thought that it would make a good tip. Are you using such parts as tips Jeremy?
Since it is quite flat, CM can be brought down nicely.  - so I thought!
I built a top and spun it on a concave mirror as base.
It spun for while, not much wobble, nice. But then it stopped spinning and did not fall over!!
Hm, CM clearly above contact point, not even a recessed tip....? What is going on? Does the curvature of the mirror keep the top from falling? No, a test on flat surface proved that that was not the problem, it did not fall on the flat surface either.
-- some thinking----

  (the head slap emoji is still missing)
 
 - Ah, it is not enough for a thing to be a top to ask that CM is always above contact point! The radius of the curvature of the contact point tip needs to be smaller than the distance from contact point to CM !! Otherwise is can't fall.  So I did not really built a top I think.
You must have discussed those things before around here, I admit I did not perform search on that now. Did you agree on rules what can be considered a spinning top for this forum at least?
But there is also some small satisfaction in that I found that out on my own. So it seems I have found a nice LEGO tip for not so low CM (I guess Jeremy will know the exact radius of that curvature off the top of his head.)

[ta0]

There is something I don't understand. The golden ratio black top needs a density radio of 3.6. But on the photos it looks like the heavy side has three layers, with the bottom layer carved out, so I would expect it to be more like 2.6 

Black isn't a golden square offset top -- just the closest I could get. Its density ratio is 0.360, not 3.6. That means that the small black rectangle is 1 / 0.360 = 2.78 times denser than the larger one.

The algebra behind the formulas was much nastier than expected. Defining the density ratio as I did simplified the math a bit, but using its reciprocal instead would have caused a lot less confusion.

Doh! (I need the face palming smiley that ortwin mentioned). Moving the decimal point one place was not a smart move! 
For the maximal offset case, the Ansatz approach makes it very easy and the result couldn't be simpler: ratio of densities equals the square of ratio of sizes.

I would call the black top a golden square top: it's closer than many things in nature that are said to have the golden ratio.

[ta0]

- Ah, it is not enough for a thing to be a top to ask that CM is always above contact point! The radius of the curvature of the contact point tip needs to be smaller than the distance from contact point to CM !! Otherwise is can't fall.  So I did not really built a top I think.
You must have discussed those things before around here, I admit I did not perform search on that now. Did you agree on rules what can be considered a spinning top for this forum at least?
But there is also some small satisfaction in that I found that out on my own. So it seems I have found a nice LEGO tip for not so low CM (I guess Jeremy will know the exact radius of that curvature off the top of his head.)

Although when analyzing the tippe top, this comes up, I don't think I ever internalized as a condition for true "topness". From now on I'll remember that having the center of mass above the tip is a necessary but not a sufficient condition for a true top. Thanks for the insight!

[ortwin]

[...
Doh! (I need the face palming smiley that ortwin mentioned). Moving the decimal point one place was not a smart move! 
For the maximal offset case, the Ansatz approach makes it very easy and the result couldn't be simpler: ratio of densities equals the square of ratio of sizes.
...

Yyyyes,... ? But you had that result in  reply #9 already?

[Jeremy McCreary]

For the maximal offset case, the Ansatz approach makes it very easy and the result couldn't be simpler: ratio of densities equals the square of ratio of sizes.

I would call the black top a golden square top: it's closer than many things in nature that are said to have the golden ratio.

Good point about so-called "golden" things. Pretty loosely applied to man-made things like buildings and works of art, too. By that standard, Iowa is golden -- at least on the map.

What do you mean by "maximal offset case"?

[ortwin]

...
Where do I need to draw that interface line to maximize the offset?
The "beer can Ansatz" we discussed earlier will hold here as well. That is why I tried to put the black circle, symbolizing the stem, on the line that separates the two materials.  ...

This is from the very first post in this topic. That is the case he refers to. The maximal offset for a given pair of densities.

[ortwin]

A LEGO top thing that has the general appearance of my beloved "Quark Top" (the grandfather of commercial endurance tops?), and how I transformed it into a real top by exchanging the tip for a black ceramic ball:

I looked up what that toy is called it reminds me of:  "Stehaufmännchen" in German, tumbler toy in English. While looking for that, I also came across this "Grand Illusions" video. Quite certainly linked somewhere in this forum already. Now I have to put that also on my Christmas/birthday wishlist:

[the Earl of Whirl]

I got all excited about that glass but then I looked it up....$62 dollars!!!

[Jeremy McCreary]

Seems like that glass is the last thing a drunk needs.

[Jeremy McCreary]

So it seems I have found a nice LEGO tip for not so low CM (I guess Jeremy will know the exact radius of that curvature off the top of his head.)

My own lingo, just to keep things straight: "Pivot" = the entire surface subject to touching the ground at tilts short of the scrape angle. "Contact" = the instantaneous physical interface between the pivot and the ground.

We use "tip" pretty loosely around here. The meaning's usually clear from context, but not always. As a modular topmaker, I tend to think of the "tip" as an assembly including the part providing the pivot and all the other parts needed to secure the pivot to the rest of the top.



Of the 5 LEGO pivots above, the 4 on the left see the most use by far. They also go by other names. Each has its pros and cons. The pivots on the Mixel joint ball and Technic ball have tiny flats at bottom dead center, but the effect on behavior is generally negligible. All 5 are otherwise spherical.

The Technic ball (red) is very handy, as it fits securely on the end of a Technic cross-axle (black) with no other mount needed. Many different parts can be used to mount the minifig mic and Mixel joint ball securely — provided you're willing to shorten the small rods attached to them.

We've discussed the profound effect pivot radius of curvature has on top behavior and contact resistance many times. When I want to maximize spin time, favor sleep, and minimize travel, I cut myself a 1.6 mm pivot off the end of a round-tipped 4L antenna. (This pivot is found on several other parts as well.)

I use the boat skid (far right) as a pivot only when I want lots of travel or certain behaviors from a high-CM top. Seems to have more resistance than all the others -- especially when translucent, and especially on concave surfaces. (The polycarbonate LEGO uses to make translucent parts has significantly higher coefficients of static and sliding friction than the ABS used for opaque parts.)

Often, my choice of pivot is dictated by how securely I can mount it on a given top and how desperate I am to lower the top's CM. Pivot wiggle is bad.

- Ah, it is not enough for a thing to be a top to ask that CM is always above contact point! The radius of the curvature of the contact point tip needs to be smaller than the distance from contact point to CM !! Otherwise is can't fall.

Great point! Hadn't quite put it together that way. So in order for a spintoy to fall, its CM must be above both the contact and the pivot's center of curvature when the spin axis is vertical.

Two tops made from weighted Spinjitzu turntables below. The purple one with the Mixel joint ball pivot falls right over at rest. The red one with the boat skid pivot stays upright. Vertical CM heights are about the same.

You must have discussed those things before around here... Did you agree on rules what can be considered a spinning top for this forum at least?

Many discussions, but still no unanimous agreement. For a spintoy to qualify as a true top, I personally require that it have a non-zero critical speed, below which it has no stability against gravity and falls at the slightest provocation. Many others share that view, though not necessarily in those words.

(the head slap emoji is still missing)

Yes, and I'm really going to need it for myself in my next post. So from now on, I'm using as the dope-slap emoji.

[ortwin]

@ta0: there are two quite different discussions going on in this topic. Only one of them is really considered with the "offset top". Do you consider to split the other one off the "offset top" topic?

...
Of the 5 LEGO pivots above, I use 4 on the left the most. ......

Thank you for sharing your top tip secrets in such detail that they can be really of use!

 

... So in order for a spintoy to fall, its CM must be above both the contact and the pivot's center of curvature when the spin axis is vertical.

...

That is a point I need to keep in mind  especially when I go for endurance tops with recessed tips. I like to use those 4 mm ceramic balls as tips, and things become even more critical with the 6 mm and 1/4 inch ceramic balls. And now that it is out, the Mistress will watch even closer!
That is an advantage of Iacopo's sharp tips, he can basically ignore the curvature and put his contact point very very close to CM.
- But wait, I am not sure it is true what I just said. Is the critical speed dependent on the distance of CM above the contact point or of the distance above "center of radius of pivot" ?? I still did not familiarize myself well enough with that critical speed calculation.

[Jeremy McCreary]

For the maximal offset case, the Ansatz approach makes it very easy and the result couldn't be simpler: ratio of densities equals the square of ratio of sizes.

Duh, sometimes I'm the denser rectangle. The "Ansatz" case forces the top's CM onto the boundary between rectangles such that y_CM = B. As you said, it then becomes very easy to get

(S - B)² / B² = γ = 1 / δ,

where S is the square's side length, B < ½ S is the small rectangle's short side, and γ > 1 is the areal density ratio (small / large rectangle, as you prefer).

This leads directly (and with many fewer steps than I took initially) to the Ansatz-specific quadratic relationship I noted above, here in terms of γ and the small rectangle's area share β = B / S:

(γ - 1)β² + 2 β - 1 = 0

Solving for β in terms of γ and vice versa for the Ansatz case gives

β = √γ / (γ - 1)
γ =  (β² - 2 β + 1) / β²

But getting y_CM in the non-Ansatz case (White top above) is still pretty tedious. In terms of β and γ now, as measured from the small rectangle's free edge parallel to the boundary,

y_CM = ½ S [(γ - 1) β² + 1] / [(γ - 1) β + 1]


Using (S - B)² / B² = γ makes it easy to hunt for LEGO-friendly Ansatz solutions. For the square LEGO plate I've been using with S = 16 studs,

Doable:
B = 5 studs, γ = 4.84
B = 6 studs, γ = 2.78
B = 7 studs, γ = 1.65

You've already seen a B = 5 top and a B = 6 (Black). Here's a smoothly spinning B = 7 top to the right of Black. The couple unbalance is minimal.



Not doable due to unreachable density ratios:
B = 1 studs, γ = 225.0
B = 2 studs, γ = 49.0
B = 3 studs, γ = 18.8
B = 4 studs, γ = 9.0

[Jeremy McCreary]

Thank you for sharing your top tip secrets in such detail that they can be really of use!
...
Is the critical speed dependent on the distance of CM above the contact point or of the distance above "center of radius of pivot" ?? I still did not familiarize myself well enough with that critical speed calculation.

You're welcome. The standard critical speed formula assumes a point contact fixed in space. It depends only on the CM-contact distance, the specific AMI, the specific TMI at the tip, and the local acceleration of gravity. Don't recall the differences WRT the versions for spherical pivots and tops (like tippe tops) with CMs below the pivot's center of curvature. But there are strong similarities among all of them.

If ever there were a formula for a topmaker to become one with, as a design heuristic if nothing else, it would be the standard formula for critical speed.

[Jeremy McCreary]

So in order for a spintoy to fall, its CM must be above both the contact and the pivot's center of curvature when the spin axis is vertical.

Not a spintoy, but I guess we all knew this at some level, if not a verbal one...