epicyclic curve

Do you have an equation?

Seems like my Tech Drawing classes at school weren't a waste after all ;-)

An epicyclic curve is the locus of a point on the perimeter of a small circle rolling (without slipping) around a larger circle. I'm not sure that there is a single mathematical expression for it - but it's one of those things that's easier to draw with a pair of compasses and a rule than on CAD.....

I think that to draw it in SW, you would have to think more like a draughtsman and 'construct' the circles and lines, then tie together key quantities using equations and linked values.

Might have a go at this one when I get a minute....

Martin

Maybe this will help in the effort...

I don't believe there is a way to draw this in SW with 8 place mathematical accuracy. But here is something that "may work":

-Two simple cylinders cooresponding to the minor and major diameters placed into an assembly with gear mates between the two and other constraints that only allow appropriate motion

-a point on the perimeter of the smaller which will allow you to extract its location

-create an assembly sketch and convert the above points' location, remove the coincident constraint and make the point fixed.

-rotate the larger cylinder

-rinse and repeat

-a spline created from the generated points should give a reasonable representation

Creating a macro to auto-rotate the large cylinder, extract perimeter point data, and repeat at a given increment should allow for a reasonably high degree of accuracy

Even better yet. You may be able to google a working model of a toy that was around when i was young ( may still be, not sure ). It was called a spyrograph (sp?). The correct model of it would already have all the constraints necessary to create the correct motion type.

"Brian" mathematical accuracy. But here is something that "may work":

With that you've got some equations. I have a macro on my website that will draw a spline for an equation. Theoretically it should be able to accept equations with periodic terms, but I've never tried one as complex as the ones on the above site.

If you want to try my macro, it's at

, follow the link for Macro Library. It's called "eqcurve".

As I remember, polar coordinates are the best way to deal with cardioid shapes.

Good luck!

Matt

Try this. Just set a and m. a is the OD of the circle around which you want an epicycloid and m is the number of cusps. Start a part. This will draw a sketch on the Front plane.

' ****************************************************************************** ' C:\DOCUME~1\kellnerp\LOCALS~1\Temp\swx2044\Macro1.swb - macro recorded on 05/24/05 by kellnerp ' ****************************************************************************** Const pi = 3.141592654

Dim swApp As Object Dim Part As Object Dim boolstatus As Boolean Dim longstatus As Long, longwarnings As Long Dim FeatureData As Object Dim Feature As Object Dim Component As Object Dim skPts() As Double

Sub EpiCycloid(ByVal N As Long, ByVal a As Double, ByVal b As Double)

ReDim skPts(N + 1, 3) As Double Dim i As Long Dim x, y, z As Double Dim phi, dphi As Double

dphi = 2 * pi / N

For i = 0 To N - 1

x = (a + b) * Cos(i * dphi) - b * Cos((a + b) / b * i * dphi) y = (a + b) * Sin(i * dphi) - b * Sin((a + b) / b * i * dphi) z = 0#

skPts(i, 0) = x skPts(i, 1) = y skPts(i, 2) = z Next i

x = (a + b) * Cos(0 * dphi) - b * Cos((a + b) / b * 0 * dphi) y = (a + b) * Sin(0 * dphi) - b * Sin((a + b) / b * 0 * dphi) z = 0#

skPts(N, 0) = x skPts(N, 1) = y skPts(N, 2) = z

End Sub

Sub main()

Set swApp = Application.SldWorks

Set Part = swApp.ActiveDoc boolstatus = Part.Extension.SelectByID("Front Plane", "PLANE", 0, 0, 0, False, 0, Nothing)

' a is the OD of the circle around which you want the epicycloid. m is the number of cusps.

a = 20# m = 6#

'Don't change anything below here.

b = a / m N = 10 * m

Call EpiCycloid(N, a, b) Part.InsertSketch2 True

Part.ClearSelection2 True

For i = N To 0 Step -1

Part.SketchSpline i, skPts(i, 0), skPts(i, 1), skPts(i, 2)

Next i

End Sub

Isn't that the equation for a face of a gear tooth (less root and crown)? If so, there maybe accurate equation based profiles around that could be modified. Bob

It would be nice if SW had a "locus" function, where a point could trace out a line as it moved according to a set of rules, equations, parameters, etc. Next up in complexity a generating function could carve away at an extrusion according to rules- think gear-shaper cutter generating a gear tooth profile.

Bill (big ideas for someone else to implement) Chernoff

That is why they call Pro/E a parametric modeler. The equations here are parameter driven. SW cannot generate a parametric curve like the little macro does. In fact the macro is pretty simple to modify to create any kind of parametric curve. Just change the equations for x and y.

This curve is not the curve for gear tooth faces. That is called an involute and there is a fairly simple construction to make it.

Wow thanks guys, I will define the problem a little more now and see what you think.

I have a wheel and it has three protrusions on it. This wheel rolls on a flat surface As it rolls, it fits thru a hole smoothly. I need the profile of this hole.

Now I came up with the easy way out. Make the protrusion the same as a tooth on a pinion. The matching shape is but a flat tooth as you would find on a rack.

This is an intresting topic, so it would be good if it is explored further.

Kind thanks for all help so far, I will try some of the ideas suggested.

Giorgis

By the way, Solidworks should have a method of drawing epicyclic curves. You can generate all manner of gears by editing the parameters. I guess I should explored the formulas used to generate the profiles.

What I have done instead is put a pinion and a matching rack. Then pinch the profiles by using "convert entities" End up with only one tooth.

It will work exactly ... priceless ?!?! I know I took the easy way out, but that is what engineering is all about

G

DAMN ... I am wrong, the thooth profile on the gears drawn by SolidWorks are actualy arcs on a circle. I should have guessed that they are pictorial. They look so close though %$#@^&#$@. Anyway, I should be able to find the exact profile onece I have figured out the correct gear to put in.

Giorgis

How is this an epicyclic curve?

I may have things confused.

Effectively I need a wheel withone tooth in it to cut a flat surface.

G

PS: I am having trouble using the macro you supplied. Could you give me some pointers ?

An involute (gear tooth) and an epicyclic curve are very similar.

Picture the following for drawing an involute: =B7A string wrapped around a spool with a pencil on the end. =B7As the string unwraps with the string held tight, the traced path is an involute.

I once needed to cut a true involute on CNC from SW geometry. I laid out a series of construction sketches (8 in total) equivalent to "unwrapping" string at 5=B0 intervals. The endpoints of the 8 "strings" were used to anchor a spline. An additional aid was that the "strings" were instantaneously perpendicular to the involute path, making it possible to use tangent constraints to help form the spline.

1. Make a dummy macro by turning record on and then off.
2. Save the macro as epi or some other descriptive name.
3. Open the macro you just created in macro editor
4. In the editor window erase everything.
5. Paste in the macro I posted
6. Start a new part.
7. Make sure the name for the Front plane matches that in the macro.
8. Hit run. The macro should draw a close approximation of an four petal epicycloid.

By the way, the units for the 'a' dimension is in meters. You may want to convert unless you need a 60+ foot epicyclic.

I thought the macro wasn't working until I thought to zoom out.

...

->Try this. Just set a and m. a is the OD of the circle around which you

->want an epicycloid and m is the number of cusps. Start a part. This

->will draw a sketch on the Front plane.

->'******************************************************************************

->' C:\DOCUME~1\kellnerp\LOCALS~1\Temp\swx2044\Macro1.swb - macro recorded on 05/24/05 by kellnerp

->'******************************************************************************

->' a is the OD of the circle around which you want the epicycloid. m is

->the number of cusps.

->a = 20#

->m = 6# ...

Good point. SW works in metres.

Well I just threw it together in 15 minutes. User friendly it ain't, but it works and it is layed out reasonably logically to the point that it can be tweaked. Any parametric functions can be inserted for x and y. Even z can be added if you want to open a 3D sketch. .

You should also be aware that at the inflection points where the epicycloid touchs the base circle the spline does not come to a perfect point, so in that region it is inaccurate. To fix it play with the indices in the point generator to get just one petal. Then the spline should be pretty accurate.