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dgfoster

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Trig Solution to Tapering Hole Diameter Problem

For anyone interested in the question I am posting the trig solution to the problem of spacing holes on a tapered arm with uniform edge-to-edge spacing along the axis of their centers. I decided to start a new thread as this trig problem was not really the main point of the first thread Is there a scratch-free layout method available? . The trig solution included is applicable to any number of holes with any taper and any spacing.

The "bottom line" solution is in red at the bottom of the page. For completeness the derivation of the solution is included above that.

There are two images attached. One is a photo of the drawing of just two circles with an exaggerated taper angle as this setup allowed a more clear separation of some of the tangent and intersecting lines than would have been possible if the circles were drawn to the same taper as the part I was making. The second picture shows the part I intend to make drawn to scale with the spacing of the holes and their radii indicated.

The work I did on this makes no sense in some ways as a nearly as good and practical result could have been obtained in a much shorter time by just using paper and pencil or a CAD program and empirically fitting the circles between the lines drawn. I guess the challenge of getting a math solution sucked me in.... Maybe this math solution will be of use to someone else who doesn't care to grind through the mental gymnastics to figure it out.
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The goal of this exercise to determine a method to calculate the diameters of circles of smoothly diminishing diameters and spaced at a specified distance from circumference to circumference and with their centers falling on a straight line. In the diagram shown only two circles are considered. But, once the calculation is made for one pair of circles, subsequent calculations can be made using the same formula.

Definitions:
a is the angle subtended by one line tangential to both of the circles and another center line through the center of the two circles.
R is the known radius of the larger circle.
S is the distance between the circles as measured on a line connecting their centers. Its length is arbitrary but is constant.
r is the radius of the smaller circle. Its value is unknown and we want to calculate it relative to R for a given S.


First we need to find the height of the line F-E:


A-B = R/cos(a)
E-D = (S+R)
D-B = [E-D/cos(a)] sin(a) = (S+R) tan(a)
A-D = A-B - D-B = F-E = [R/cos(a)] - [(S+R) tan(a)]
F-E = [R/cos(a)] - [(S+R) tan(a)]


Now that we have calculated F-E we can find the radius (r) of the smaller circle by determining F-E in terms of r:


H-G = H-J + J-G = r sin(a) + J-G = r sin(a) + r = r[sin(a) + 1]
H-E = H-G/cos(a)
E-G = H-E sin(a) = {r[sin(a) + 1] sin(a)}/cos(a) = r[sin(a) + 1]tan(a)
F-E = E-G + r = r([sin(a) + 1]tan(a) + 1)


So we have now determined F-E in terms of R and S and also determinedF-E in terms of r.


So: F-E = [R/cos(a)] - [(S+R) tan(a)] = r([sin(a) + 1] tan(a) + 1)
r = {[R/cos(a)] - [(S+R) tan(a)]}/({[sin(a) + 1] sin(a)} + 1)


r= [R/cos(a)] - [(S+R) tan(a)]
.......{[sin(a) + 1] tan(a)} + 1

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Last edited by dgfoster; 12-28-2013 at 09:33 PM. Reason: formatting

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12-29-2013, 12:11 AM #2

Metalcutter

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All I can say, is I'll keep you DGfoster, for any wild math I need help with. I'm not to bad in trig, but I'm
Stuck on formulas for an equal angular spiral, and the formulas for transformation from log-polar to Cartesian coordinates. Also known as a logarithmic spiral. I'd like to have an understandable formula where I could spread sheet it and have it geometrically accurate where I could plug in an angle variable. I can build the excel sheet.

If thats ok,
Stan-