Calculating circle spacing on tapered strips with 1 inch cutter

[DannyH]

I'm looking to find someone who can help me with my problem...

My problem is that I need to remove material from linear tapered strips with a 1 inch round cutter. The circles need to connect to each other at half the section depth over the entire length of the strip

--->

http://www.keone.com/hollow.gif


Because the diameter of the strips decreases the circles need to be placed closer to eatchother so that the circles keep connecting eatchother at the hart of the strip.

My problem is how to calculate the distances...

Can someone please help me out with this?

The picture in the link will make it more clear

Thanks a lot!

Danny

 

[HackaB]

I don't know if there is a general formula, but if you fix the first circle at a certain point along the strip, you can calculate the distances between the successive circles iteratively. Take a look at the image I attached. Fixing the first circle defines the angle $$\theta_1$$. That first intersection is the only one you have to measure. If R is the radius of the circles (I guess .5" in your case?), then the distance between circles 1 and 2 is

$$d_{12} = 2 R \cos \theta_1$$

To find $$\theta_2$$, you can use the known slope "m" of the half-depth line. Just compute "rise over run" from the first intersection (circles 1 and 2) to the second (circles 2 and 3):

$$m = \frac{\Delta y}{\Delta x} = \frac{R\sin \theta_1 - R\sin \theta_2}{R\cos \theta_1 + R\cos \theta_2} = \frac{\sin \theta_1 - \sqrt{1 - \cos^2 \theta_2}}{\cos \theta_1 + \cos \theta_2}$$

Since $$\theta_1$$ is known, you can solve this quadratic equation for $$\cos \theta_2$$. Then the distance between circles 2 and 3 is

$$d_{23} = 2R \cos \theta_2$$

You can repeat this process to get the successive distances. This seems awfully tedious though. Maybe someone else will be inspired to come up with a better way.