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I Need a help in solving an equation (probably differentiation

  1. Nov 20, 2017 #1
    I am trying to find out the interference condition between tool and a part. The below attached snapshot is the equation between interference and machine feed. At dy/dx = 0, I will have max. interference, which I intend to find. Except x and y every alphanumeric character in the following equations is a constant.

    I tried to get it in y = f(x) format, however because the constants and y are so complexly attached, I failed. It would be of great help, if someone can help me find out an equation that gives me the dy/dx = 0 condition. Thanks.
    cpu8N.jpg

    https://i.stack.imgur.com/cpu8N.jpg
     
  2. jcsd
  3. Nov 20, 2017 #2
    If ##y=f(x)## you have ##x=f^{-1}(y)##, if the inverse ##f^{-1}## exist. As is shown at the Wikipedia page, you then have
    $$f'(x)=\frac{1}{x'(y)}$$.
     
  4. Nov 20, 2017 #3
    Let's see...
    I would go for x=f(y)
    So dx/dy = -d(G10+G6)/dy = -tan - d(G10)/dy
    Since we are going to be looking for a denominator that goes toh zero, we can drop the tangent term.
    So we are looking for 1/(d(G10)/dy)=0.

    Attacking the G10 equation:
    Let A = D9/H; So G10 = A sqrt( (H/2)^2 - (D12 + Rtip -Sqrt(Rtip^2-y^2) )^2 )
    Let B = (H/2)^2 - (D12 + Rtip -Sqrt(Rtip^2-y^2) )^2; So G10 = A sqrt(B)
    the derivative of that is:
    d(G10)/dy = (A/(2 sqrt(B)) ) (dB/dy) = (A/(2 sqrt(B)) ) ( -(D12+Rtip-sqrt(Rtip^2-y^2))^2 /dy)
    Since we are looking for excursions to infinity, the factors "A" and (-(D12+Rtip-sqrt(Rtip^2-y^2))^2) will not assist and can be dropped.
    So we are looking for 2sqrt(B) = 0; thus B = 0
    So (H/2)^2 = (D12 + Rtip -Sqrt(Rtip^2-y^2) )^2
    H = +/-2(D12 + Rtip - sqrt(Rtip^2-y^2) )

    Specific information about whats positive and negative helps here.
    So I'll leave the rest to you.

    BTW: Check everything - I can make mistakes.
     
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