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Creating a Step-off Chart for Turning Radii

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Kim:
Over in my Pennsy Switcher thread Dave asked about how I was setting up my step-off sheets for turning curves and such.  And Perertha posted some of the ways he was doing the same thing.  So I decided to start this thread to help keep the discussion together.  Over there it keeps getting separated by the normal build stuff.   This should make it a better reference for people as the discussion from everyone will be in one place.

Pertertha had one sheet for ellipses that looked really nice.  Iíll let him post that here.  It seems like a very good way to do simple curves when youíre using a tool with a sharp point (like a parting tool).  Iíve done that before many times too.

However, for my example, Iím going to post the spreadsheet I used for the steam dome saddle.  It has some additional complexities like using a round-nosed tool and requiring an X and Y offset for the curve.

I tend to create a new sheet for each curve Iím trying to make a chart for.  If itís an easy one, it's easy so it doesnít much matter.  If it's hard, itís probably hard for a reason I havenít dealt with yet.  It seems difficult to make a general spreadsheet that covers all cases. However, as I do this more, I may change my mind! :)

The curve itself is fairly simple.  Itís just an 11/32Ē radius curve on the outer edge of the part.  It happens to be set back from the top of the part by 3/32Ē also.  Thereís a little flange that provides registration for the sandbox dome.  Hereís a diagram of the basic shape.  It also labels the main dimensions used for the calculations.

Rc = Radius of Curve in the part
Rt = Radius of the tip of the tool
Xoffc = X offset of the Curve from the center line of the part
Zoffc = Z offset of the curve from the front of the part
Theta = Angle of the cut of the curve

This picture also shows where I set the origin for the DRO (labeled Part Origin).  I set the X-axis using the outside diameter of the part. The Z axis is set to Zero at the face of the part.  I set the CENTER of the tool here. I considered using the left side of the tool since that was a little easier to do, but I did that and then offset it by the radius of the tool.  I felt that it made the calculations simpler to do this.  Or at least, it was easier for me to visualize the whole thing that way.  :)

Another thing to note is that the angle of the tangent point of the tool with the part will be the same as the angle of the curve on the part.  If that doesnít make sense, look at the drawing and move the tool tip along that curve in your mind.  You should notice that the purple arrow (the angle of contact with the tool) and the red arrow (the angle along the curve on the part) are always the same.  That simplifies things a lot.

Now letís look at the spreadsheet.  Iíve uploaded the full thing below if youíre interested, but Iíll include pictures of it to help explain what Iím talking about as we go.

I always begin by making the curve I want to end up with.  In this case, that is the columns labeled Xc and Zc (columns A and B).  Column C, labeled delta, is the Z step size that I chose for that step.  You can see that the delta changes as you walk through the chart.  I made this one based on the Z step size and calculated the X value for the curve using Pythagoreanís theorem.  I have the angle, so sin/cos would have worked fine too.  But I find Pythagoreanís theorem to be easier to think about when I can get away with it. These columns define an arc of radius Rc with its origin at the center of that arc.



Now, I didnít start with 50+ rows.  I usually start with 10-20 to get the math worked out right then add rows and adjust the deltaís till I get the step sizes that Iím comfortable with.

Since Iím using the round-nosed tool, I needed to calculate the angle of that point in the curve so that I could use that angle to also calculate the location of the tangent point on the cutting tool.  Thatís where columns D and E come in.  D is Theta, in radians. The spreadsheet thinks in radians, so I use that here. But I tend to think in degrees, so I converted radians to degrees in column E to help me.  These columns represent the location of the tangent point relative to the center of the circle of the cutting tool.



And then with the angle calculated, I calculated Xt and Zt (columns F & G), the locations of the cutting point of the tool based on that angle.  I used Sin() and Cos() for this.  These values are relative to the center of the cutting toolís radius.  In this diagram, the circle represents the round-head tool being used.  It may not be a full circle like this shows - mine is just a half-circle, but it works great.  When I ground it, I was aiming for a 5/64" diameter (0.0781") tip.  I missed a little and it came out about 0.076", but that was close enough for my work.  So, in my case, Rt= 0.038, as you can see at the top of the spreadsheet section of the pic.



Now Iíve got most of the info I need to figure out the actual coordinates for the cut.

Columns H & I are labeled CutX and CutZ. This defines the point where the cut actually occurs but in the coordinate space of the part.  Up till now, Iíve used a couple of coordinate systems; one for the curve (Zc, Xc), one for the tool (Zt, Xt), and now going to start converting them so that they are relative to the part (the Part Origin).  To do this we have to add the X and Z offsets (Xoffc and Zoffc) to the curve values (Zc and Xc).  This puts the curve in the right place on the part relative to the part origin.



CutZ, CutX is where we want the cut, but we still need to include the offset due to the rounded tool.  As we said, this offset changes based on where we are in the curve. That is where the TTipX and TTipZ come in (columns J and K).   These are the coordinates at the tool tip that will position the round-nosed tool where we want it.  Thatís easy to do because we already calculated the offset of the cutting point from the tool tip; Xt and Zt. However, we have to take into account that the origin for these offsets is the center of the radius of the tool.  Including the Z off set (Zt) is straight forward. But for X we have to take the difference of the offset from the radius as shown here.



Weíre almost there!  I have my DRO set to read out in diameter, not Radius. So I have to double the Z coordinate since all of the Z dimensions have been in radius to this point.  And for X, I just have to make it negative, since thatís the minus direction on the DRO. This probably isnít that big of a deal. Sometimes I donít bother with that step. I just know itís going to be negative and I make that adjustment in my mind.  But in this spreadsheet, I did it.  These values are in columns L and M and are named Xdro and Zdro.

Zdro = 0 - TTipZ
Xdro = TTipX * 2

The ĒZActCutĒ and ďXActCutĒ are not that interesting.  Itís just me backing out the radius tool offsets and DRO mapping to make sure I could get back to the right Ďcutí position.  This was an error-checking/validation step for me.  It took me a while to get everything straight in my mind and these error-checking columns helped me find my math problems and verify things.

The Xdelta and Zdelta (columns calculate the difference in the X and Z coordinates between two adjacent steps. I used this when adjusting the size of the delta in column C and in deciding when Iíd added enough steps to the table.  The MinDelta column just does a MIN() function on the X and Z deltas.  As long as one of them meets my acceptable minimum step, Iím good.  I used some conditional formatting on these columns just to make it easier to spot outliers in the step size.


The graph is also a verification/trouble shooting aid. It helps me see if the arc is going to come out the way Iím thinking.  And in this case, I plotted two lines; the blue one is the actual cut line (CutZ, CutX) showing the final radius being turned, and the red one is the position of the tool tip (TTipZ, TTipX). The tool tip also inscribes an arc but is a section of an ellipse, not a circle.    The two curves only coincide at one position; the very bottom of the cut. Then the cut starts to become offset from the tip more and more as you progress around the arc until the cut is 90o from the tip, on the very edge of the tool. I flipped the direction of the axis around in the graph so that the curve would show up in the same orientation as Iíve been thinking about it. Not completely necessary, but nice.


For actual use, I bolded the Xdro and  Zdro columns for easier reading and hid all the columns that I didn't need for the lathe operation.  Actually, I didnít hide them because that messes up the graph (the graph wonít show hidden columns). So I just made the width of all the columns really narrow (like one or two pixels) except for the ones I care about.  Then I printed it out for use in the shop.

In this case, I started from the bottom of the chart and worked my way up. I'd set the X position and move Z from the end of the part to the specified Zdro coordinate.

Sorry, this was so long-winded. But hopefully, it was somewhat interesting and/or helpful.  If I left out details you're interested in, ask.  If I've provided too much detail, sorry, just ignore my post!  :Lol:

Thanks, 
Kim

Dave Otto:
Hi Kim

Thanks so much for the effort that you put into this. I have been busy and only have had time to glance over it. I need some time to digest this and see if I can understand it all.
I'm sure that I will have some questions.

Dave

petertha:
Kim asked for my prior post to be put here

I made an Excel table to generate a stepover routine for an elliptical shape. An ellipse is handy because you can define the X & Y segment lengths independently, resulting in a wide range of aspect ratios to suite the purpose. For example elongated like a bullet or stubby like the end of a propane tank. (Or if a & b are equal, it becomes a circular section). Ellipses are nice smooth shapes & the intercept will be tangent to the straight shaft segment.

Hopefully the table example & sketch makes sense. Red are input values. Parameters a & b define the X & Y segment lengths respectively. X (L to R) means the step the cutting tool from left to right, but practically you probably want to use X (R to L) Right to Left. My DRO displays diameter when I infeed so I also put Y-dia that on the table. So you make the step-overs using something like a parting blade, blue the resultant stair step surface, then finish down with file & paper until no more blue is showing.

The underlying equation is : y = [ (1- x^2/a^2) * b^2 ] ^ 0.5     Y-dia = 2 * Y


Hopefully I didn't make make any math errors, if so please correct me. Apologies for the Kozo build interruption.

petertha:
Kim> That's a nice clean way to do it Petertha.  Very slick spreadsheet, thanks for sharing it.  I like the general case of the ellipse. That's an elegant way to make a very general spreadsheet for cutting curves.
I have often found I need for an X or Y offset too, for various reasons.  Another wrinkle is using a round nosed tool to cut the profile, rather than a sharp point.  That adds another level to the calculations.  But sometimes it's really nice to use a round tool rather than a sharp one - like when I was doing the parabola for my headlight reflector.  That was a convex shpae that would have been harder to file the stairsteps out of. Using a round nosed tool left a very nice finish on the reflector and made it worth the extra math in the spreadsheet.

Peter> You have raised some good points. If one wants to use a larger radius cutting tool, the spreadsheet math gets more complicated because the cutting tool nose radius is both coincident to a coordinate on the curve and also tangent to the curve at that point. The other issue is how to do the step offset. If you do equal X increments & calculate corresponding Y, that makes for more straightforward manual machining, however a likely byproduct will be larger steps to interpolate with the file as a function of the curve shape. To assist, one could subdivide that particular area with finer steps using same spreadsheet methodology. Or (like sketch example) divide the curve into equal segments, but now you have oddball X and Y coordinates to position, so choose your poison.

Again, sorry for the geometry rambling. The only reason it was on the forefront of my brain is I was thinking about a circular ended lifter acting on a cam profile. Its a similar problem. Hats off to the guys that figured this stuff out hundreds of years ago without CAD.

Kim:
Thanks Petertha!  :ThumbsUp: :)

Kim

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