Calculating the Radii of Ukulele and Guitar Arches

 I have finished writing my book "Left-Brain Lutherie:  Using Physics and Engineering Concepts for Building Guitar Family Instruments:  An Introductory Guide to Their Practical Application".  Details can be found here.       

It's fairly routine to read about an arch of so many feet radius for the top of a guitar and another for a different number of feet radius for the back of the guitar, but how do you go about measuring these radii? That is, if you wanted to make a form for a router to make a dish two feet wide with a radius of 10 or 15 feet how do you go about estimating the shape of the curve?

        Where to look? Probably any beginning geometry text would have sufficient information, but I found a proper figure on page 398 of the CRC Standard Mathematical Tables, Chemical Rubber Publishing Company, Cleveland, OH 1959.

        I don't claim that the following analysis is the most elegant, but it works. I'd be happy to hear from others regarding such calculations and be willing to offer their analyses as well. Figure 1 shows the arch cross-section.

        The arch under consideration has a number of components : 1) the arc of the arch shown as the solid line ABC; 2) the linear width of the arch, solid line ADC (also termed a chord); 3) the radius of the arch, solid lines AO, BDO and r; the height of the arch, BD. Note that angles BDC and ODC are right angles and that OBCO is an equilateral triangle.

        For our first case, let's try to find out the radius of the back of a classical guitar. The width of the lower bout (ADC) is 14" and by placing a straightedge gently on the top of the arched back we can measure a distance of 1/4" between the edges of the back and the edge of the ruler (BD). So we know the value of BD (1/4") and CD (7") and want to know r.

Figure 2.  Guitar Side View

So,

OD² = r² - CD²

so,

r = BD + (r² - CD²) ^1/2

rearranging and solving for r,

r = (BD² + CD²)/(2 x BD)

Thus for our example,

r = (.25² + 7²)/(2 x .25)

= 98" = 8.17'

        Now let's try a different case. Suppose that we want to create an arc that will guide a router to cut a dish shape into a form for arching the top of a guitar to a radius of 20'. If the dish will be say, 18" wide then we need an arc probably 24" wide. So if ADC is 24" then CD is 12". r is 20' which is equal to 240". At this point, we need to solve for BD using CD and r.

Figure 3.  Dish Side View

From above,

BD = r - OD

and

OD² = r² - CD²

so,

BD = r - (r² - CD²)^1/2

Therefore for this second example,

BD = 240" - (240² - 12²)^1/2

= 0.30"

The name of the spreadsheet for the two problems described above is ArcCalcs.xls and is an Excel spreadsheet.

ArcCalcs.xls
(to download double-click or option-click)

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