pattern equation

#1
For those that give a hoot, here is the equation for only one of a million patterns that can come from a sine or cosine wave on a pattern bar.

x=t - rsin(atan(-sin(t)))
y=cos(t) + rcos(atan(-sin(t))), where t is the evaluation parameter.

This one's easy because there is a nice mathematical function that describes the pattern bar and the solution is analytical and not numeric. This equation is adaptable to other continuous functions as I think all that might need to change is the substituion of the first derivative (slope) for the "-sin(t)" factor which is the slope of the cosine curve. Also those with better math skills and determination can carry on with other patterns that are not so nicely described and for ones on a rose engine. Have at it! I suspect Bill Ooms has some nice stuff in his program.

I'll go back to making chips, now.

Cheers,
Rich
 

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#3
It's not based on it - it's just an example (" ...Below is an example of the path of the straight line engine cutter when using a rubber with a radius equal to the half amplitude of the cosine wave...") to give additional context to the discussion. The red line is just another example. However, it just so happens that for the sine/cosine wave the half amplitude radius is a critical number where if the rubber is larger than that you'll start decreasing the result amplitude. It's nothing earthshaking and it's pretty obvious with other patterns that a rubber whose radius is bigger than the smallest radius of curvature you will lose, or not reach, part of the pattern. Still, just an example.
 
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#5
This is an excerp from an email I wrote to John Edwards. They are my words and don't include any of his so I have no trouble publishing it. If my recollection is correct I had a discussion on this topic some time back with Rich L. who put the formula in much more elegant mathematical terms, but I was raised doing farm work.

As for Ramsay's lead screw on the pattern bar tower the use of this formula would reveal that if making pattern bars that the 12TPI does not lend itself poorly to the production of simple, concave pattern bars that will produce most of the basic patterns including that that require the sine wave to accomplish.

I love the discussion here and have only two regrets. One is that I'm up against a couple of hard deadlines which are approaching all to quickly at a moment when I lack proper motivation. The other regret is that I feel a bit behind the curve in aptitude for this being that I've put the tool to the work all too little.

I'm interested in what you guys think about what I've written below. I'd love to see a treatise drawn up that would incorporate both the "farm" approach and the mathematical formulae as well.

David


I spent a lot of time boiling down some of these patterns to a repeatable formula that could be used on any rose engine. I then simply used that formula on mine. No doubt many before me came up with the same formula; indeed, Rich Littlestone seemed to be on top of it to a greater level by far than me in regard to straight line work. However, besides Daniels no one has really come close to putting it out there for the general populous. Why would they make it easier for someone to come along and compete? Today we no longer really compete.

For instance, here’s what I came up with for the basket weave.

N=Number of lines per run
A=Amplitude
P=Phase (in degrees for rose work)
Period=distance peak to peak in degrees (Perhaps “frequency” or some other word would be better here, but I don’t know that word)

Slide Movement= A
N

P= 1/2 Period
N

Example

24 bump rosette with 21 thou amplitude and runs of three
Period 360/24=15 degrees divided by 2 =7.5 degrees divided by 3 =2.5 degrees
The crossing plate will be phased each time by 2.5 degrees or very, very close it. -----P=2.5
In the case of a crossing notch with 144 you would phase I notch(two with 288 as on my machine).

Amplitude .021” divided by 3 =.007” Slide movement =.007” which can be ascertained quite quickly with a dial indicator and the sectors set.



If you wanted runs of 4 it would be

Slide movement = .021”divided by 4= .005”
Phase=7.5 degrees divided by 4=1.875 degrees phase with each cut.

I have 192 on my crossing plate which would be one notch (200 will also work well as it’s close enough).


Do you see any fundamental flaws?
 
#6
David, you explained this to me when visiting in August and I haven't had a chance to experiment with it. I think the math is good. I think it needs some diagrams to help it out for anyone who isn't comfortable with visualising the math.

I think it would be a good idea to create a series of "recipes" for patterns that people can reference. I believe that most of them can be broken down to a principle formula that can be applied to other bars with similar characteristics. At the very least it will give people a starting point to work from.
 
#7
I need a picture or a sketch! Please, just to make sure.

This is diagonal basketweave, right? Sharp corner patterns with coincident cuts.

The arithmetic looks OK except I would do one simple thing in practice --

If my rosette frequency did not match any crossing plate notch then I'd fashion up a simple paper protractor scale (and maybe a vernier scale, too) and glue it to the either the crossing plate or the spindle or the main wheel depending on space and how the engine is set up. That would give precise repeatablilty and would not be "off" a bit because of "off period" crossing plate notches. Or, ... super glue a notch pair, triplet, ... on the crossing plate to give me the notches exactly at the spaces I need them. This would be just an eighth inch thick piece with notches for the pawl.

Chris, you can work on the polar equivalent of the sine wave exercise... (smiley face) starting place r=d/2+a(ƒ(12Θ) for an 8 inch 12-lobe rosette of amplitude a. (more smiley faces!) I like pulling your chain!

Cheers,
Rich
 
#8
I'll see about writing this up with sketches so Rich can follow along ;)

Ok Rich you're going to need to start giving us math lessons. All the symbols you're using are vaguely familiar, but it's been twenty years since I've needed to use them. Maybe dumb it down for the slow kids in the class and break down what each symbol means.
 
#9
I was experimenting with this pattern a bit today. I'd like to propose one small change that I think will improve it. I think that slide movement should be (A+X)/N where A is the amplitude, N is the number of lines per run, and X is a small amount added to the amplitude. Daniels suggests that when cutting a barley corn, you add A/5 between the rows (4th ed, pg#378 ). This gives the cuts a bit of breathing room. I'm not sure that A/5 is always the right number, but I think there needs to be a bit of space (I usually create enough space between the center line of the cuts so the right edge of the left hand cut meets the left edge of the right hand cut).

If you think of this basket weave as a barley corn with a few additional cuts in between, the added space makes sense. I found that when cutting the pattern it improved the overall look of it. Can someone else give this a try and let us know what you think?

I've got a diagram mostly finished explaining the cut sequence. Hope to have it up tomorrow.
 
#10
How 'bout a new thread called "cutting barleycorn" or something? This thread's gotten all mixed up. IMHO. Then another thread called "cutting diagonal basketweave." With pictures!


Cheers,
Rich
 
#11
Ok I'll separate them out. And add pictures. Trying to get some weather proofing done on the shop before it gets too cold. Only so many days of above freezing temps to work in…
 
#12
David
I'm just catching up with you guys on this, having been diverted from the shop in the last couple of months. I like your "farm" arithmetic in your email of October 10th above (where you quoted your message to John Edwards). In it you reference a series of 288 notches on your machine; specifically you said that "In the case of a crossing notch with 144 you would phase I notch(two with 288 as on my machine)."

I assume you aren't referring to the Lindow rose engine but, if you are, are you referring to a different crossing plate than I have on my Lindow? Absent these series of notches presumably one would just crank the worm over 2.5 degrees. Notches would be easier of course, especially when moving 1.875 degrees (as in your second example). I guess in that latter case one would need to use the 200 series of notches (1.80 degrees) on the Lindow.

Can you please clarify if you have an alternative crossing plate for the Lindow or were you referring to a different machine?
Ramsay
 
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