Theoretical Surface Roughness — Turning

A round tool nose fed along a rotating bar traces a helix, and between one pass of the helix and the next it leaves a scallop. The height of that scallop is what this formula gives: $R_t = f^{2}/(8 r_\varepsilon)$, a piece of circle geometry with no metallurgy in it whatever.

The square is the useful part. Halving the feed quarters the roughness, while doubling the nose radius only halves it. Feed is the powerful lever and the cheap one, and that is why a finishing pass is defined by its light feed rather than by anything else. It also runs the other way: a heavy roughing feed leaves a surface four times worse for every doubling, which is why roughing and finishing are separate operations rather than a single compromise.

Now the honesty, because this is the formula most often quoted as though it were a prediction. It is a floor, not a forecast, and the real surface is usually worse — often by a factor of two or more. What the geometry omits is everything that actually happens at the cutting edge. Built-up edge: at low speeds in ductile materials, workpiece metal pressure-welds onto the rake face, grows, breaks off and takes surface with it, and it dominates the finish completely when it is present. Vibration and chatter: any relative movement between tool and work writes itself onto the surface, and a slender part or a long boring bar will always be the limiting factor rather than the feed. Tool wear: the nose radius in the formula is the one the insert had when it was new. And ploughing: below a feed comparable to the edge radius of the tool, the edge stops cutting and starts pushing, which puts a hard floor under the whole exercise — past a certain point, feeding lighter stops improving the finish and starts making it worse.

One more trap, and it fails good parts. This is $R_t$, the peak-to-valley height, and your surface tester almost certainly reports $R_a$, the arithmetic mean deviation. For this particular scalloped profile $R_a$ runs near a quarter of $R_t$, so quoting an $R_t$ figure against an $R_a$ specification makes a perfectly acceptable surface look like a failure by a factor of four. Check which one the drawing calls for before doing anything about it.

The way around the whole trade-off is the wiper insert, which carries a small straight or large-radius section trailing the nose. It flattens the scallops the main radius leaves, giving roughly the finish of half the feed at the feed you actually programmed — often doubling the feed rate at constant surface quality. It asks for a rigid setup in return, because a wiper's longer contact pushes harder.