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.
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!
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.
So why do cam design people call functions that are a simple sin wave or at least the simple harmonic motions a Cycloid?
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
Kinematics with MicroStation Ch07E Cycloidal Cams - YouTube
I'm trying to figure out a "Cycloidal Acceleration" Curve... Just need to make sure that I'm not being completely stupid.
*), 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.
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.
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?