Winding technology.

[MichaelG]

Speed

Thanks for the wxMaxima link - Brilliant engineering tool - I was doing it by hand : never again?

Went to http://calc.matthen.com which in an online access to the Maxima and substituted in


acos((a^2+b^2-(d+v*x)^2)/(2*a*b))

for differential (replaced C1 with d and used x for time)

I do not find the extra v you were talking about
I also am having trouble with the dimensional verification of the speed - thinking that I am missing a radian conversion factor somewhere

Graphs are coming

 

[MichaelG]

Graphs

As Peter said the Position is smooth and continuous
The Speed is continuous but not smooth
with a spike at the end points

Interesting the system is not symmetrical in position, I think this occurs due to the angle to the extruder reaching 90 deg before 45 of motion


x Axis is mm of Feed
y1 is Deg
y2 is Deg/s

 

[Peter Nachtwey]

There wasn't an extra v, I was wrong.

I tried to edit my post but I couldn't access the forum and I had a meeting to go to. I wanted to add this:

eq10: (c0-v*t)**2=a*a+b*b-2*a*b*cos(x);
eq11: solve(eq10,x);
eq12: 'diff(x,t)=diff(rhs(eq11[1]),t);
grind(%)$

'diff(x,t,1) = (t*v^2/(a*b)-c0*v/(a*b))/sqrt(1-(t^2*v^2/(2*a*b)-c0*t*v/(a*b)+c0^2/(2*a*b)-b/(2*a)-a/(2*b))^2)$

You should be able to cut and past the first four lines to get the result. grind(%) just converts the pretty print to text for pasting in forums.

wxMaxima doesn't simplify very well.
x is the angle
c is the length of the wire from the extruder to the corner of the winder. This isn't really the correct answer though. It assumes we know v(t) and it is coming from a reference encoder and may change. What we really need is (dx/dc) and (dc/dt) and use the chain rule. The motion controller should have the chain rule built into its cubic interpolation. The cam table really needs just x as function of c. The problem is that the cubic spine interpolators will not handle the non-smooth changes in dx/dt nicely. A cubic spline interpolator will need to have some extra tricks. Hopefully Knut's winder is going so slow it won't be a problem.

Thanks for the wxMaxima link - Brilliant engineering tool - I was doing it by hand : never again?

I always use wxMaxima or Mathcad. Mathcad is prettier but not as powerful as wxMaxima. Mathcad also cost a lot of money.

Michael, if you are going to use wxMaxima you should find all the examples you can and download all the tutorials. I do math like this all the time so I have taken the time to learn wxMaxima pretty well. There are a bunch of little gotchas that you must learn.

 

[Honga]

Smells like my formula!

 

[knutabru]

Hiding under my desk....

I tried to edit my post but I couldn't access the forum and I had a meeting to go to. I wanted to add this:

eq10: (c0-v*t)**2=a*a+b*b-2*a*b*cos(x);
eq11: solve(eq10,x);
eq12: 'diff(x,t)=diff(rhs(eq11[1]),t);
grind(%)$
 
'diff(x,t,1) = (t*v^2/(a*b)-c0*v/(a*b))/sqrt(1-(t^2*v^2/(2*a*b)-c0*t*v/(a*b)+c0^2/(2*a*b)-b/(2*a)-a/(2*b))^2)$

You should be able to cut and past the first four lines to get the result. grind(%) just converts the pretty print to text for pasting in forums.

wxMaxima doesn't simplify very well.
x is the angle
c is the length of the wire from the extruder to the corner of the winder. This isn't really the correct answer though. It assumes we know v(t) and it is coming from a reference encoder and may change. What we really need is (dx/dc) and (dc/dt) and use the chain rule. The motion controller should have the chain rule built into its cubic interpolation. The cam table really needs just x as function of c. The problem is that the cubic spine interpolators will not handle the non-smooth changes in dx/dt nicely. A cubic spline interpolator will need to have some extra tricks. Hopefully Knut's winder is going so slow it won't be a problem.


I always use wxMaxima or Mathcad. Mathcad is prettier but not as powerful as wxMaxima. Mathcad also cost a lot of money.

Michael, if you are going to use wxMaxima you should find all the examples you can and download all the tutorials. I do math like this all the time so I have taken the time to learn wxMaxima pretty well. There are a bunch of little gotchas that you must learn.

Hello.

It seems I have bitten over more than I could chew with this project. I had not seen the complexity in this before the Motion-Masters started to dig into this problem. I must say that I am deeply impressed in how you guys delve into the problem and showing me, and others, the issues in using a square drum in a winder.

MichaelG made an drawing in Excel that pretty much sums up the setup of the winder, so you all have that as a good reference to the hardware. The drum will have 2 meters sides, but the length to the extruder or the traversing-apparatus are not known yet.

I have concluded so far that a Technology-CPU S7-315T must be at leased used in order to achieve the motion of the drum.
There is a matter of using a cam or a "non-contact" dancer. I would prefer to use a cam as a non-contact analogue measuring-frame for a 1 mm diam. thread, could be difficult to find. Using a cam in a S7-315T has 2 methods, either as an expression or as an interpolation table. If the velocity-profile has the for of scallops, then a interpolation table will not do the trick. I then remains with using an expression. The technology CPU has this formula for building a cam:


I also attach a Small PDF regarding the cam-formula
View attachment Polynom.pdf

This may look like I'm asking for my "homework" to be done, but if I could get an formula that fit in the Technology-expression or I could acquire a non contact measuring of the slack in the thread, I'm sure I would be on my tracks again for this project.

I will check if it's possible to superimpose cams and use interpolation tables in conjunction with an expression in order to achieve the correct cam if a single expression are not enough.

Regards from
Knut

 

[Peter Nachtwey]

Smells like my formula!

I didn't see an acos() in your formula.

If the velocity-profile has the for of scallops, then a interpolation table will not do the trick.

The interpolation table may be the best choice because any method will have a discontinuity at the corners. I would program in 4 cams, one for each side. I would make the cam table go a little past the corners of the winders. When starting to pull on a new corner I would switch to the new cam on-the-fly. Hopefully the winder is turning slow enough that the sudden change in acceleration doesn't cause a problem with the tension.

I don't know how well that polynomial formula will approximate the acos() function. I see the warning about only relying on the third order terms and below. Try the 4 cam table methods first. Hopefully your controller will allow changing cam table on-the-fly and have a way of going from one spline to another nicely like ours can.

I must say that I am deeply impressed in how you guys delve into the problem and showing me, and others, the issues in using a square drum in a winder.

It is called experience. Also, wxMaxima makes doing the math easier. I knew right off this was going to be 'fun'. wxMaxima can also expand the acos function I did it but the results look messy with imaginary numbers.

if I simplify the term inside the acos(angle(c)) and just call substitute x for angle(c) then the series is

taylor(%pi-acos(x),x,0,6);
grind(%);

x:=angle(c);
%pi/2+x+x^3/6+3*x^5/40$
This approximation would have to plot that against the exact solution and Siemens warns about the x^5/40 term not being accurate.

 

[PLC]

What was the result?
How did this project go?

 

[knutabru]

Hello.

Well.... the project was given to a company that also was able to build the winding machine and thus beeing able to deliver a more complete sollution.

I have see the machine and the winding speed seems to be solved using a small non-contact measuring frame. And since the speed is very low it seems to be adequate

But on the positive side, I've got to learn and know more about electronic cams in motion control.