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Math Help - Find radius to 3 arcs tangent to each other

  1. #1
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    Find radius to 3 arcs tangent to each other

    Hi

    I am trying to find equation for the radius for the arcs in the drawing(all arcs have equal radius). I also require an equation to work out half the chord length of the first arc. The height of the arc are fixed at 120mm. The first and and third arcs are both tangent to the second arc.

    I am trying to machine this pattern on a CNC and require it parametric. So the machine will work out the pattern if the length changes.

    Thanks in advance

    Liam
    Attached Thumbnails Attached Thumbnails Find radius to 3 arcs tangent to each other-image2993.png  
    Last edited by bigz; April 22nd 2012 at 05:05 PM.
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  2. #2
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    Re: Find radius to 3 arcs tangent to each other

    Quote Originally Posted by bigz View Post
    Hi

    I am trying to find equation for the radius for the arcs in the drawing(all arcs have equal radius). I also require an equation to work out half the chord length of the first arc. The height of the arc are fixed at 120mm. The first and and third arcs are both tangent to the second arc.

    I am trying to machine this pattern on a CNC and require it parametric. So the machine will work out the pattern if the length changes.

    Thanks in advance

    Liam
    Not sure wether I understand your question correctly or not ...

    1. I've modified your sketch a little bit so I get 3 symmetrically arranged arcs. Unfortunately the most important measure is missing: The "height" from the red straight line to the "top" of the arc. I've taken 60 mm.

    2. Then you are dealing with a right triangle: See attachment #1

    3. Using Euklid's height theorem (and your measured values) I get:

    60(2r-60)=\left(\frac{585.977}2 \right)^2~\implies~\boxed{r \approx 745.352}

    See attachment #2
    Attached Thumbnails Attached Thumbnails Find radius to 3 arcs tangent to each other-krsbogenwelle.png   Find radius to 3 arcs tangent to each other-euklid_theor3.png  
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  3. #3
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    Re: Find radius to 3 arcs tangent to each other

    Hi earboth. Unfortunately my first drawing was a bit confusing. Hopefully this ones better. I am trying to find 1/2 chord length for the first arc and the radius of the arcs. All I know is the length of the 3 arcs at any given time and the height. All arcs are equal radius. The 2 end arcs are tangent with the middle arc and tangent with the top construction line. The middle arc is tangent to both end arcs and tangent to the base construction line.

    Thanks

    LiamFind radius to 3 arcs tangent to each other-curve-door.png
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