Hyperboloid of revolution twirler

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tornitore45
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Hyperboloid of revolution twirler

Post by tornitore45 » Sat May 11, 2019 1:55 pm

I want to build one of these.
https://www.youtube.com/watch?v=cfrQSzDfFoI

Anybody has plans or formulas to cut the Hyperbola given the rod mounting attitude and location?
Mauro Gaetano
in Austin TX

Russ Hanscom
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Re: Hyperboloid of revolution twirler

Post by Russ Hanscom » Sat May 11, 2019 2:19 pm

Do an internet search, lots of examples there.

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tornitore45
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Re: Hyperboloid of revolution twirler

Post by tornitore45 » Sat May 11, 2019 5:38 pm

I am aware of the examples.
I am looking for dimensioned plans OR an analytical way to relate the position and inclination of the revolving "stick" to the projection on the plane-on-axis where the hyperbolic slits are cut.
Mauro Gaetano
in Austin TX

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Bill Shields
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Re: Hyperboloid of revolution twirler

Post by Bill Shields » Sun May 12, 2019 8:04 am

Dimensioned plans are difficult to follow since the form is..by definition...

Write a macro that does the math and feed it to a CNC mill and you will get what you want.
Too many things going on to bother listing them.

Russ Hanscom
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Re: Hyperboloid of revolution twirler

Post by Russ Hanscom » Mon May 13, 2019 8:59 pm

Correct, you only have to do one quadrant and then mirror to get the other three.

After looking at the project and reviewing math definitions, I am pretty sure what is shown is a parabolic curve, not hyperbola - at least that is the equation I would start with.

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Re: Hyperboloid of revolution twirler

Post by pete » Mon May 13, 2019 11:51 pm

Milled slots done in metal with a thou or two of clearance then yeah a mathematical formula for the shape and a good multi axis cnc mill might be needed. But looking at that video and the clearances he's got, any stockmaker that's used to fitting custom gun stocks to barreled actions could do the exact same, or maybe even a lot closer fit with some lamp black on the shaft and many many trials and just removing the wood where the lamp black touches as the shaft moves further towards the parts vertical C/L. Do one end as Russ mentioned, flip the pattern for the other end on the same side, and there's your exact pattern for the opposite side. A bearing on a router bit could cut the whole shape by just having a half pattern if your planning to do it in wood. Very tedious to get that one half of the pattern I'll admit, but not really that hard to do at all. Zero complex math involved. :-)

I know a guy in the U.K. who I bet could use the same method and do it in metal with nothing more than a drill, some dykem and his hand files.

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tornitore45
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Re: Hyperboloid of revolution twirler

Post by tornitore45 » Tue May 14, 2019 10:55 am

All good points but no CNC here, only dials.
I was a bit lazy but being in vacation near Buffalo (I know, who would vacation in Buffalo) between the never ending rain and the cows in the pasture I got down and worked out the geometry.

The slits are definitively an Hyperbole an not a Parabola, that was know a priory given the construction definition.
Choosing the parameters is a matter of taste but an tall X skinny in the waist leads to unexciting, fairly straight, lines. To give prominence to the curved region of the Hyperbole the slits fit in a more square rectangle.

I can drill a series of holes at the coordinates calculated in the spreadsheet attached in PDF form.
Hyperbola-Twirler.pdf
(300.23 KiB) Downloaded 36 times
Mauro Gaetano
in Austin TX

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Bill Shields
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Re: Hyperboloid of revolution twirler

Post by Bill Shields » Tue May 14, 2019 11:47 am

so drill a lot of holes and attack it with a file or a band saw
Too many things going on to bother listing them.

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mklotz
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Re: Hyperboloid of revolution twirler

Post by mklotz » Tue May 14, 2019 12:23 pm

The Cartesian equation of a hyperboloid is:

(x^2 + y^2)/a^2 - z^2/b^2 = 1

With the origin set at the center of the figure, define the x direction to be left/right and the z direction up/down. The y direction is then perpendicular to the board. To obtain the equation of the cuts in the board, take y to be zero, i.e., the equation in the plane of the board.

What results is the Cartesian equation of an hyperbola.

x^2/a^2 - z^2/b^2 = 1

When z = 0 , corresponding to a horizontal line through the center of the board, we have x = a, which determines the separation of the two curved lines forming the hyperpola.

For z nonzero, we have:

x^2 = a^2 * (1 + z^2/b^2)

so, as z increases the separation between the +x value and the -x value becomes greater and greater thus creating the characteristic flare of the hyperbola.
Regards, Marv

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Bill Shields
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Re: Hyperboloid of revolution twirler

Post by Bill Shields » Tue May 14, 2019 12:31 pm

you can feed this equation to a printer, have it print out a 2D line (curve)

glue the paper to your material and use the 'prick punch jig borer' method to spot hole centers.

Note: be sure you are printing 1:1 scale !
Too many things going on to bother listing them.

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tornitore45
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Re: Hyperboloid of revolution twirler

Post by tornitore45 » Tue May 14, 2019 4:17 pm

The OP was not about drawing, printing, machining an Hyperbola.
The OP was: Given the position of the stick on the twirler rotating disk, what is the equation of the Hyperbola to cut the slit so that the stick pass through?
Reference to a contraption like: https://www.youtube.com/watch?v=cfrQSzDfFoI
An Hyperboloid can be described in many ways. The mode of interest here is the surface described by a line rotating around an axis NOT parallel to the line.
The definition includes the distance Line-to-Axis and the a\Angle between the line and the plane containing the axis and the distance segment. An additional specification impose a finite length of the rotating line, but such is only a practical consideration.

One can build or visualize a model by threading strings between the teeth of two parallel gears (so to speak).
Chose an offset between corresponding teeth and lay strings inclined both ways.

This type of definition does not immediately yield the equation of the intersection of the Hyperboloid wit a plane through the axis.

My original problem was to go from the location (Radius and Inclination) of the stick and the coordinates of the intersection with the plane.
I did not derive the Cartesian equation but derived the Parametric equation x and z in function of omega, the angle of rotation of the base holding the stick.

I found more convenient to convert the specs from the construction point of view relative to the rotating base to geometrically more significant specs as distance and inclination of the stick respect the axis.

I posted a PDF of the spreadsheet used to work out the answer. The formulas are hidden but I was surprised how simple they turned out.

Once I have the coordinates of many points is just a matter of drilling and filing. The desk size model will be made of metal, the slit as narrow as practical and hand crank operated.
Mauro Gaetano
in Austin TX

Inspector
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Re: Hyperboloid of revolution twirler

Post by Inspector » Tue May 14, 2019 5:33 pm

Here is one is done as a clock that is pretty cool.
https://www.youtube.com/watch?v=DgqTGXB9KNk

Pete

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