Winding technology.

[Peter Nachtwey]

Hint, cosine law.

 

[leitmotif]

Peter
I assumed a square 4" on side and one RPM. Solved for single revolution.
I did it using tangent. The problem I found using tangent is you can only do it it 45 degree "chunks", tan of 90 is undefined so I added total periphery of a 45 chunk to preceding 45 chunk. Can only give tabular data.

y = 0.044x linear regression is R2 = 0.9997
Not much help I admit for accurate motion control.

See att'd file both chart 1 and sheet "tangent".

I tried using cosine combined with sin works fine until 45 but over that it all falls apart could be done but I think would have to do in 45 chunks.
This attempt is in sheet "sin cos"
Dan Bentler

 

[Peter Nachtwey]

PThis attempt is in sheet "sin cos"
Dan Bentler

What, no graphs?

 

[Peter Nachtwey]

Start with the cosine law

I provided step 1
http://en.wikipedia.org/wiki/Law_of_cosines
In the picture in the link assume that 'b' is the distance from the center of the winder to a corner of the the square spool. 'a' is the the the length of the extruded material and 'c' is the distance between the source of the extruded material and the center of the winder. The rate of change in the length 'c' is the extruder speed. What is the rate of change in α as a function of the extruder speed and the angle α. Finding the derivative is the second step.

 

[leitmotif]

What, no graphs?

Yeah there was a graph in the file. Look at "grahp 1". Pretty straight line with small spikes every 45 degree.

Dan Bentler

 

[Peter Nachtwey]

I found the graph but I don't see the spikes. A few comments would help. I don't see the rate calculations or the cosine law.

 

[leitmotif]

Peter

What I calculated was the periphery - I just discovered one tiny error - I did this while stationary - now I gotta make it turn and redo calcs. Let me study law of cosines and learn how to use it.

Was a good review of trig. Thank God I still remember Suzy can tell.

Dan Bentler

 

[504bloke]

Peter

What I calculated was the periphery - I just discovered one tiny error - I did this while stationary - now I gotta make it turn and redo calcs. Let me study law of cosines and learn how to use it.

Was a good review of trig. Thank God I still remember Suzy can tell.

Dan Bentler

? Huh ?

 

[L D[AR2P#0.0]]

knutabru - Please provide a drawing showing the path of your thread before the winding drum and also indicate the true shape of the "square" winding drum.

 

[Honga]

I found length from extruder to barrel edge to be:
where d = width of barrel.
and L = length from extruder to barrel center.

a = (root(2)*d*cos(alpha) - L)i + root(2)dsin(alpha)j

Given alpha = wt

a(t) = (root(2)*d*cos(wt) - L)i + root(2)dsin(wt)j

Then:
v(t) = (-w*sin(wt)*root(2)*d)i + wcos(wt)*root(2)*d*j

From this you could find the magnitude and %variation from peak and use it to scale your w.

Interesting to note that distance to barrel center has no effect on velocity. This leads me to believe the above is incorrect.

Another required step is to find out the limits for which wt is true. You would do this by finding the angle of a when it is equal to 90 - Arg(root(2)d(cos(alpha)i * sin(alpha)j))

 

[Peter Nachtwey]

Interesting to note that distance to barrel center has no effect on velocity. This leads me to believe the above is incorrect.

You didn't use the cosine law. Also you are computing the velocity of the wire given the rotation of the square spool when you want to compute the rotation rate of the square spool as a function of the velocity of the extruded material and the angle of the square spool.

Another required step is to find out the limits for which wt is true. You would do this by finding the angle of a when it is equal to 90 - Arg(root(2)d(cos(alpha)i * sin(alpha)j))

That is the forth step.

 

[Honga]

Y = length of extrude to barrel (easy to figure out)
Y' = V_extrusion
L = distance of barrel center to extrusion
r = 1/2 width of barrel

YY'/(root(2)Lr) = wsin(wt)

LHS are all known values. RHS has multiple solutions eliminated by calculating the range.

 

[MichaelG]

My attempt

My attempt This uses Cosine rule
- probably have the velocity wrong (had to look up the derivative)

Also in this application we are not looking at the winding angle onto the drum - ie If my pictures are looking from the side then we are not worrying about looking from above

The pdf is for those who do not have Office 2010 - the conversion munched my circles and squares into ovals and diamonds

Zip file contains Excel xlsx as only xls files are allowed to be attached


View attachment Square Winder.pdf

 

[MichaelG]

Errors seen after a good night sleep
1) Velocity is negative
2) in the Start and finish angle calculations The Sqrt(2) should divided not multiplied

probably comes from not trying to get a graph (working on that)


The traverse axis removes all worries about looking from above
I think someone mentioned using current/torque control - problem with this is the effective radius is changing constantly so the tension in the thread will changing

 

[Peter Nachtwey]

You are very close and on the right track.

Good show so far. You have a problem with a sign. My numerator would be c1+v*t where c1 is the initial length at t=0. I think you have an extra v in the numerator. Also we disagree on some signs under the square root.
I still need plot this. I haven't spent much time on this to work it out completely. I use wxMaxima to do the calculus.