Finding the radius

Or, are all radius(es) calculated the same!? Okay, I just finished doing some repair work on a very old Harmony Vita Uke. And what a truly wonderful little uke it is. The look might not be for everyone, but the sound is fantastic, and the intonation is better than almost any uke I've met. So, naturally I've taken every measurement I can to keep a file on this model. I'd like to draw up a set of plans for a future build. And I have looked to see if any exist, but didn't find any.
No matter, I enjoy drawing them up on the computer, which naturally helps with the accuracy. But here's the thing; I'm measuring the back and the top to find the radius of each (the vita uke has a full radiused top F/B/L/R). So I started by laying a ruler across the top and back and measuring the gap at the ends. Then I made a scale drawing on the computer and matched a radius to the profile. And here's where it gets interesting. The radius that matched up to the top is 8 feet!, and the back matches up to a 4-foot radius!!. And I hope Im not miscalculating this... I know the radius of a circle is from the center point to to the perimeter. (15 foot radius makes a 30 foot diameter circle. Right?) So then I did some drawings with a standard 15 foot radius and put it on a soprano size uke profile, it's barely noticeable, but when I put a 15 foot diameter arc on it, it looks like the right amount of arch. So what gives? Is everyone calculating the radius the same? Here's my drawing of the profile of the Vita Uke body. the red line represents an 8 foot radius arc and the green line is the 4 foot radius. The blue spheres represent the measured gap (.125" for the top, .25" for the back). Whadda you think about this?



PS I know Pete Howlett built one of these... Pete, what numbers did you find for your build?

What makes you think the plates are built with/to a spherical radius?

Well, I don't know, what other kinds or radii are there?

Lots. Have a think about how many ways you can create an arch. More bend in one direction that the other.

For instance, more across the grain than length wise. Or more along the grain than across. It's quite common to see both those scenarios.

Then how about a parabola? Or a dome in part of the plate but flat in the other? Or a bent plate like on some styles of instruments. Then you can have ones that will have a flat or even tiny bit of recurve in them on the perimeter.

Our friendly English gent TimBuck has something on this here:

http://ukulele-innovation.tripod.com/soprano-arched-back-s.html

Beau Hannam Ukuleles said:

Our friendly English gent TimBuck has something on this here:

http://ukulele-innovation.tripod.com/soprano-arched-back-s.html

Click to expand...

Well that's cool to see he measured a rather small radius of 5'2". Perhaps the 4' I measured on the Vita Uke is not that far off. And I have no problem measuring the uke in multiple directions i.e. with the grain, or across, I just want to make sure my method is correct. I had not come across a 4 foot radius before.

That's most likely because most everything you will read about radius on a plate trickles down from the guitar world.

I've worked a bit with tighter radii than most guitar guys use, mostly trying to get a flat mandolin plate to support a normal height mandolin bridge. After building a 9' spherical dish I went to a 3' cylindrical bracing platform. A cylindrical radius is arched across the grain (in my case) but flat with the grain. As the spherical arch gets tighter, fitting the neck/bridge relationship becomes more problematic. Fitting a neck to a cylindrical arch is about like fitting one to a flat plate, only the bottom of the bridge must be arched to compensate for the arch of the plate. The geometry is the same.

On a soprano uke a tighter spherical radius gives more support to the whole system and looks cool up to a point. The small measurements involved prevent the neck/bridge relationship from getting too messed up. Tighter back radii present few problems. But tighter means stiffer, and eventually compensation would have to be made in the rest of the system, such as lighter bracing. Either way you arch, a flat landing pad must be provided on the top for a fingerboard extension.
On the whole, I don't see a reason to get too carried away, though. If conventional arching works for you, there's probably not too much reason to chase the ghost of unknown improvements.

No flat necessary, John, just cantilever the fingerboard extension the way Howe Ormes were done 115 years ago... As far as I know, they were the first to incorporate the cylindrical arch, but in their guitars, not mandolins.

jcalkin said:

I've worked a bit with tighter radii than most guitar guys use, mostly trying to get a flat mandolin plate to support a normal height mandolin bridge. After building a 9' spherical dish I went to a 3' cylindrical bracing platform. A cylindrical radius is arched across the grain (in my case) but flat with the grain. As the spherical arch gets tighter, fitting the neck/bridge relationship becomes more problematic. Fitting a neck to a cylindrical arch is about like fitting one to a flat plate, only the bottom of the bridge must be arched to compensate for the arch of the plate. The geometry is the same.

Click to expand...

Good points John. Those are the reasons I use a cylindrical arch for my tops (spherical for backs though.)

Rick Turner said:

No flat necessary, John, just cantilever the fingerboard extension the way Howe Ormes were done 115 years ago... As far as I know, they were the first to incorporate the cylindrical arch, but in their guitars, not mandolins.

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Rick, I cantilevered the fingerboard extensions on my mandos. They were of 1/4" thick Honduras rosewood, but I noticed a real falloff of volume and sustain when played on the floating section. Do you support the cantilever somehow on your instruments? I'm pretty sure it was the flexibility issue that ate up the quality of the sound. (And its not like I was ever hotshot enough to ever play music up there, but things have to be tested.)