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Differential equation resembling to cycloid

  1. Nov 15, 2012 #1
    What is the function corresponding to this ODE:

    In complex notation it obviously shows up like this:

    a * z''(t) + b * |z'(t)| * z'(t) + c = 0;

    The numerical solution shows a graph resembling to a cycloid.

    Thanks for any help!
  2. jcsd
  3. Nov 15, 2012 #2


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    It does? I don't see how.
    If you divide through by the surd and subtract the 1st eqn from the second, I believe you get something integrable.
  4. Nov 16, 2012 #3
    Thank you for your comment. I tried to devide and subtract. The problem is the term in the middle: (y' - eps*x') vs. (x' + eps*y')
    It makes the situation even worse - I did not succeed in finding a simplified pattern.

    The complex notation in my first post has been derived by simply multiplying the 2nd equation by i and adding the result to the first equation.

    What I investigated in the meanwhile:

    The cycloid ODE in complex notation should be

    a * z''(t) + b * z'(t) + c = 0;

    The only difference is the multiplication with |z'| in the middle which in fact produces a value near 1 for curtate cycloids with r1 << r0 (the point tracing out the curve is inside the circle, which rolls on a line AND it is close to the center).

    The ODEs in my first posts describe a phugoid, a more general form of the cycloid I suppose.

    It seems that the phugoid has no analytic solution. Any suggestions?

  5. Nov 16, 2012 #4


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    Sorry, I overlooked what happens to the RHS. My original suggestion was nonsense.
    Ah yes, I see it now. Sorry for the noise.
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