intersection point of two askew cylinders 1

[Thunderbird336]

Hello all, I have a family of parts that typically has two cylinders which intersect; I am looking for the point that is farthest from the midpoint of the largest cylinder's axis...

The angle between the two cylinders is either 45° or 60°
One cylinder, let's call it the main cylinder, is something like Ø20" and 30" long
The smaller one is something like Ø3" and arbitrarily, the same length
The smaller cylinder is translated up something like 6" and fore or aft something like 2".

We have a very small number of CAD licenses here so I am often unable to find this number when I need it to write a CNC program, what I need the number for is a starting point from which to mill a pocket. The work piece is set upon a rotary table with the midpoint of the larger cylinder's axis coincident with the rotary table center. The rotary table is then rotated to align the small cylinder's axis parallel with the horizontal machining center's spindle, or Z axis. The Y axis represents the upward translation (from the large cylinder axis) and the X axis represents the lateral translation (from the large cylinder's axial midpoint, same as rotary table center) of the small cylinder. It may be much simpler to do this in two steps though, figure out the intersection point before translating the small cylinder laterally; then I can easily use right angle trig to calculate the X and Z offset. I made two sketches of this in jpg format, the full solid and a cross section, I have (attempted) to attach them.

I know some Calculus, but not nearly enough to figure this out; it's beyond something that I can calculate with the kind of trigonometry that I know, it seems like spherical calculations. I have been unable to even find a good solid example of the standard form for a cylinder. I know what the standard form of a circle is and can use that for finding the intersection point(s) of two circles and hoped to find something similar for a cylinder. It would be helpful if I could program a calculator or make an Excel spreadsheet to get this coordinate for my programs for the (frequent) times when I need this point and cannot obtain a network license for our 3D software.

Thanks in advance,
-Gary

 

[MikeHalloran]

... and all of that is why I used Rhino.

I've done it on the board, and I've done it in AutoCAD, and Rhino is an order of magnitude easier and faster. Solidworks can probably do it too, but Rhino is faster and much cheaper.







Mike Halloran
Pembroke Pines, FL, USA

 

[prex]

The solution can be derived from the article pointed out by cowski. Using the solution in cylindrical coordinates, the point sought is the one with maximum z: deriving the expression of z with respect to [θ], one ends, if I'm not in error, with a quartic equation in sin[θ] that's algebraically solvable (not a trascendental equation), so theoretically a closed form formula can be derived. However this formula would be so complex that it would be easier to solve the problem numerically (not to speak about a CAD solution).
To be noted that the farthest point is not in the plane of the section shown by Thunderbird336 and of course that the intersection is not an ellipse.

prex
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[IDS]

Does the attached spreadsheet do what you want?

Input (grey shaded on the Cylinders sheet) is the radius and length of the cylinders. vertical offset of the small cylinder, and longitudinal offset of the IP with the CL, and the angle. You can then adjust the level at which you want the intersection coordinates.

The XY coordinates are displayed in blue, and plotted on a horizontal plane, similar to your second sketch in the original post.

Note that I haven't checked anything.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[Thunderbird336]

The solution can be derived from the article pointed out by cowski. Using the solution in cylindrical coordinates, the point sought is the one with maximum z: deriving the expression of z with respect to θ, one ends, if I'm not in error, with a quartic equation in sinθ that's algebraically solvable (not a trascendental equation), so theoretically a closed form formula can be derived. However this formula would be so complex that it would be easier to solve the problem numerically (not to speak about a CAD solution).
To be noted that the farthest point is not in the plane of the section shown by Thunderbird336 and of course that the intersection is not an ellipse.

prex, seems like you understand this the best, it really is not simple at all. You can get in the ballpark figuring in a plane, but it is still pretty far off, like 1/2" or so depending upon how far up the larger cylinder the smaller one intersects...

This is a similar exposition...

Link

I am just not sharp enough to put them to use, he makes mention of Cartesian equation...

A curve in the xy-plane has an implicit Cartesian equation of the form
m(x, y) = 0.

and I don't even know what that is, so I cannot get there from here... but I will look over the article that cowski linked. Thanks all for so much help on this.

 

[cowski]

IDS (Doug),
The spreadsheet looks pretty good; I only verified the one example case in my CAD software. I think the spreadsheet with the use of Excel's solver add-in will get the points of interest. If I may suggest one (unnecessary) change, it would be to list the valid range for the IP Z coordinate input. It can range from offset 1 - small cylinder radius to offset 1 + small cylinder radius (unless the offset + the small radius > the large radius). This would give the user an indication of valid values for input and would also let you add constraints to the input for the solver (works without them, but would help capture intent).

Thunderbird336 (and others who are interested),
You can use Excel's solver to search for a minimum value of Small 1 Y (cell C22), by changing the value in IP Z coordinate (cell B14), then search for a maximum value of Small 2 Y (cell E22) by changing the value in IP Z coordinate (cell B14). I've only verified 1 test case, but I think the strategy used in the spreadsheet is sound.

www.nxjournaling.com

 

[Thunderbird336]

Does the attached spreadsheet do what you want?

I will check this out tomorrow, I apologize... I did not see this until I got home and read cowski's reply to you.

Gary

 

[IDS]

prex, seems like you understand this the best, it really is not simple at all. You can get in the ballpark figuring in a plane, but it is still pretty far off, like 1/2" or so depending upon how far up the larger cylinder the smaller one intersects...

The spreadsheet calculates the coordinates of the intersection points on any specified plane, so you can adjust the level of the plane, using the Excel solver, to maximise or minimise whatever you want.

I have added a calculation of the distance of the intersection points from the mid-point of the large cylinder axis, and set up the solver to maximise the larger dimension.

If that's not what you wanted, please let me know.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[IDS]

Thunderbird336 - Saw your reply to my first message immediately after posting the second one



Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[Thunderbird336]

Doug,

That is an amazing spreadsheet! I am often astounded by what people can do with Excel.

The calculation seems to break down once the small cylinder is translated up past the point of intersection with the edge of the large cylinder... I attached a screen shot of the one I am currently working on. Just for clarification, as I said before, when I can get a CAD license (such as now!) I can get these numbers out of our 3D software. Actually, to be really honest, I only know how to get close even with it, but once all of the design engineers arrive, I have no hope of getting a software license until the next day. (Just in case your wondering why I want to calculate it apart from 3D CAD software)

Is it possible to tweak the calculation to allow for the small cylinder to go higher, as in my picture? If you cannot see the picture I will upload it to the Engineering.com server, it's on Dropshots presently and, if I did it properly, is in a public folder.

Thanks so much!
-Gary

 

[rb1957]

say your pic ... now i'm confused ... if you CAD can draw it, surely you can extract the co-ordinates of the intersection ... no?

Quando Omni Flunkus Moritati

 

[cowski]

The new pic looks more like the intersection of a rounded rectangle with a cylinder. Is this a change to the requirements?

www.nxjournaling.com

 

[Thunderbird336]

say your pic ... now i'm confused ... if you CAD can draw it, surely you can extract the co-ordinates of the intersection ... no?

I can only get a CAD license early in the morning before the design engineers arrive as I said in my post to Doug...

Just for clarification, as I said before, when I can get a CAD license (such as now!) I can get these numbers out of our 3D software. Actually, to be really honest, I only know how to get close even with it, but once all of the design engineers arrive, I have no hope of getting a software license until the next day. (Just in case your wondering why I want to calculate it apart from 3D CAD software)

The new pic looks more like the intersection of a rounded rectangle with a cylinder. Is this a change to the requirements?

No, cowski, this is the same problem, I thought it would be easier to describe as a cylinder, after all, the fillet edged of the pocket is a cylinder in fact and it will always be the furthest point out where the two cylinders intersect, so it is all that matters in this case.

-Gary

 

[cowski]

Is it possible to tweak the calculation to allow for the small cylinder to go higher, as in my picture?

Wouldn't this just be changing the small cylinder z offset? Or am I missing something?

www.nxjournaling.com

 

[Thunderbird336]

Wouldn't this just be changing the small cylinder z offset? Or am I missing something?

As I said to Doug (above) "The calculation seems to break down once the small cylinder is translated up past the point of intersection with the edge of the large cylinder..." the spreadsheet will not give a result once the small cylinder goes high enough that the point of intersection is "inside" the large cylinder.

I put some numbers in the spreadsheet and kept moving up on the Z, this is as far as I can get...
Small Large
Cylinder radius 1.5 16.875
Length 40 40
Offset1 (Z small cylinder CL) 4.284
Offset2 Y IP on CL 2
Angle to Y axis -30 0 degrees

beyond this, it will give an error, the same thing will happen if you shorten the small cylinder to, say, 30"

There is still a point of intersection, it is just no longer at the edge.

-Gary

 

[cowski]

Make both cylinders longer and increase the offset 2 value until you get edge crossings. You may have to translate the output values back to return usable results for your particular problem at hand.

www.nxjournaling.com

 

[chicopee]

I have done your problem with ACAD and the attachment is in PDF format. I did not quiet follow your instructions as to the relative positions of both cylinders but they are intersecting at 60 d from the vetical Z axis. The center point of the small cylinder is intersecting a vertical line drawn between the centers of two radial lines (in x direction)drawn on the ends of the large cylinder. I made both cylinders the same length for convenience. The UCS is located on the center point of the large cylinder's base and the table values represent the x,y,z points of both holes on the surface of the large cylinder. From the data, you can select those points needed for your operation.

 

[Thunderbird336]

I have done your problem with ACAD and the attachment is in PDF format.

Very cool! I actually forgot that AutoCAD can do 3D, I have seen it done and I have walked through a couple of tutorials but it's been a few years. I have never been denied an AutoCAD license, so if I can figure it out, that would work, thanks for looking at this and helping me see that I do have resources available to do this in.

-Gary