Vertical Misalignment and Taper
This is a brief discussion of how taper can be induced by a vertically misaligned lathe as the cutting tool travels up or down relative to the center-of-rotation(COR) as one moves longitudinally along the workpiece. There is a bit of math, but not much.
I guess one advantage of being a newbie is that you question just about everything. So, when I read in the PM manual that vertical alignment was not “nearly as critical as Center-to-Center alignment” I wasn’t exactly sure what that meant. Was it something that was a little bit non-critical and should be evaluated and adjusted or a lot non-critical and could be ignored?
I wasn’t able to find any information on this but surely it should be easy to figure out. I made some sketches and I was pretty sure that using the Pythagorean Theorem would be the key to solving the problem. As some of you may recall the Pythagorean Theorem just says that if you have a triangle that has a right angle in it and you know the length of two of the sides you can calculate the length of the third. I knew two of the sides of the triangle(the drop of the tool was one, and the radius of the work piece the other) I should be able to find the third side(the new radius of the workpiece). This will make more sense in a little bit when you see some diagrams. It seemed to work, but eventually I realized that the change of the depth of the cut depended on the radius of what I was cutting. For example, if you are turning a piece that is 12” in diameter and the tool drops down an eighth of an inch as you travel along the workpiece that will have much less of an impact on the depth of cut than if the workpiece was 1” in diameter. As the tool travels down the lathe longitudinally and the tool descends, the 1” workpiece surface will “move away” from the tool faster and the depth of cut will be less because of its greater curvature. Makes sense, but how do I account for this in the calculations? Well, I needed someone smarter than myself so I consulted Dr Craig Jackson, the Chairman of the Dept. of Mathematics at Ohio Wesleyan University and told him what I was after. Fortunately, he worked as a machinist while in graduate school and got to work on it and solved it in no time. This is his note to me(I wouldn’t spend a lot of time trying to figure it out):
Here is my translation:
The bottom line is that the change in depth of the cut is equal to the distance the tool descends(or ascends) squared divided by two times the radius.
He also created a nomogram!
To use the nomogram go to the right side vertical axis and select the radius of the workpiece and follow the curved line down to where it intersects a line extended upward from the horizontal axis that represents the vertical error(wear), then follow that line over to the left vertical axis to get depth of cut error. For example, for a 1” radius piece with a vertical error of .03 gives you a depth of cut error of 0.0004. Pretty neat, thanks Dr Jackson. Now I know that the vertical error of my lathe, which is somewhere near 0.0005 is nothing to worry about. Will I use this often, I doubt it but now I can read the statement, ”vertical error is not nearly as critical as center-to-center alignment” with new understanding.
How does a descending or ascending tool effect taper? I think I will just attach a couple of drawings and let you think about it a bit. In the drawings the tool travels from a to b(the result is the same if you go from b to a), COR is the center of rotation and the “x” marks where the tool is at the level of the COG:
Well, that’s it for leveling.