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Math Help - Another Cycloid problem, attempting to describe a graphic proof.

  1. #1
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    Another Cycloid problem, attempting to describe a graphic proof.

    Another Cycloid problem, attempting to describe a graphic proof.-cycloid-motion-curve.jpg
    Okay, have a look at that picture I drew...

    What you are looking at is a reproduction of a diagram on a blueprint for a VERY old Jaguar XK over head cam design (I can't post).

    I provided what I thought were prudent dimensions... everything else... you should be able to assume. If not please ask. (okay the circle... to make drawing the plot lines easier, I used the right side for the 30 and the left side for the 15+30 increments)

    The description on the blueprint is "cycloidal motion curve"


    Take a look at my older thread...
    Integration of a cycloid.

    and you will see what I have been working on.

    I have to ask... is this diagram I posted... a graphical proof of one of the integrals for a cycloid?

    It looks a lot like the 1st integral plot... but also looks a lot like the first part of the 2nd integral plot.

    I would love some insight to this!
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  2. #2
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    Re: Another Cycloid problem, attempting to describe a graphic proof.

    It's not one of the integrals, no. The left half is a graph of x = t - sin(t), half scale.

    I.e. x = t/2; y = (t - sin(t))/2.



    Not sure where that gets you.
    Thanks from Heloman
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    Re: Another Cycloid problem, attempting to describe a graphic proof.

    Don't ya love it when you can't sit back and see something simple for what it is. Thanks . I would have never got it.
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    Re: Another Cycloid problem, attempting to describe a graphic proof.

    Ya know... On one hand Im not going to feel too bad about this, because it came right off a Jaguar engine blueprint calling it a cycloid. Maybe that's why british cars never seem to run well.
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    Re: Another Cycloid problem, attempting to describe a graphic proof.

    So why do cam design people call functions that are a simple sin wave or at least the simple harmonic motions a Cycloid?

    Examples:

    I am looking at a figure in "Cam Design and Manufacturing Handbook, second edition, Robert L. Norton" Page 134 Figure 6-8 Half-Cycloidal functions for use on a rise segment. The "a" plot does not look like a cycloid... or is it?
    Cam Design and Manufacturing Handbook - Robert L. Norton - Google Books



    http://nptel.iitm.ac.in/courses/Webc...cams/cam15.htm



    I'm trying to figure out a "Cycloidal Acceleration" Curve... Just need to make sure that I'm not being completely stupid.
    Last edited by Heloman; September 4th 2012 at 01:27 PM.
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    Re: Another Cycloid problem, attempting to describe a graphic proof.

    Quote Originally Posted by Heloman View Post
    So why do cam design people call functions that are a simple sin wave or at least the simple harmonic motions a Cycloid?
    Ah, right, so they do use a simple sin wave as the acceleration curve (*), and they call the resulting displacement curve (but not the acceleration) cycloidal because it projects from a circle as per your diagram up top, or from the vertical component of the corresponding vertically-traced cycloid, said component isolated by the groove machine in that video.

    Which, I take it, would all be gratuitous if it didn't provide a neat graphical method of fitting one complete cycle into the required rectangle.
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    Re: Another Cycloid problem, attempting to describe a graphic proof.

    Well... the only thing I have to go off of is a technical report from the late 60's with 1 sentence in it saying "The cam is characterized by its dual-frequency cycloidal acceleration corners."

    I took that sentence to mean that the acceleration plot is cycloidal. It could just as well mean that the acceleration corners are of a dual frequency cycloidal form.... thus the simple equations.

    Another Cycloid problem, attempting to describe a graphic proof.-2012-09-07-10.20.23.jpg

    See if a picture really is worth 1000 words...
    P1 to P2 is what I have been trying to define... either its based on a parametric cycloid acceleration... or... its based on a sin wave acceleration.

    The next part of the game has been trying to figure out how to get acceleration curve 1 and and de-acceleration curve 2 to have equal slopes at P2 and P3 so that they flow into eachother.

    Where we left off in the other thread I had figured out how to make Acceleration curve 1 and 2 identical, with P1(x1,y1) and P4 (x4,y4) being defined inputs.

    But what I am dealing with right now is trying to figure out the "dual-frequency" part of "...cycloidal acceleration corners".

    If in fact it turns out that these acceleration curves are just sin waves... That cam book should have what I need to figure it out.

    Either way... I feel a lot smarter than when I started this project. I guess what it comes down to now is run a analysis to try and figure out which of the two is the more efficient. Do you think the parametric based curve is even viable?
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