CNC Turning, Carbide Insert Nose Geometry & Surface Metrology (ISO 4287)
Lathe Turning Surface Roughness Ra & Rz Nose-Radius Formula
Calculate Schmaltz Kinematic Peak-to-Valley Roughness R_z (um), Exact Parabolic Arithmetic Average Roughness R_a (um), Finish Cut Cycle Time (s), and ISO N-Grade Finish Check.
Roughness scales with Feed SQUARED divided by Nose Radius—so doubling feed quadruples Ra, while a 1.2 mm nose cuts Ra by a third!
As a single-point lathe turning insert of corner nose radius R_e (mm) advances axially by feed per revolution f_n (mm/rev, for f_n < R_e), its circular tip carves helical scalloped cusps of Peak-to-Valley Kinematic Height R_z = ((f_n^2) / (8 * R_e)) * 1,000 micrometers (um)! Integrating absolute profile deviation around the mean line of that parabolic arc yields Exact Theoretical Arithmetic Average Roughness R_a = ((f_n^2) / (18 * SQRT(3) * R_e)) * 1,000 ~= (0.03208 * (f_n^2) / R_e) * 1,000 um (~R_z / 3.90)! Because R_a varies with f_n^2, halving feed cuts productivity in half to gain a 4x mirror finish—whereas switching from a 0.8 mm CNMG-08 insert to a 1.2 mm CNMG-12 (or wiper geometry) improves finish at ZERO cycle-time penalty (provided thin-walled shafts do not chatter under higher radial thrust)!
Verify on a 150 mm Hydraulic Shaft Bearing Journal Finish Pass—Feed_Per_Rev f_n = 0.20 mm/rev (B2), Insert_Nose_Radius R_e = 0.80 mm (B3), Turned_Length L = 150.0 mm (B4), and Spindle_Speed = 1,200.0 RPM (B5): Kinematic Peak-to-Valley R_z equals (0.04 / 6.4) * 1,000 = 6.25 um (E2), Arithmetic Average R_a hits 1.60 um (ISO N7 bearing-seat spec!, E3), Pass Cutting Time takes 37.50 seconds (E4), and E5 confirms "MEETS Ra <= 1.6 um BEARING SPEC"!
| CNC Turning Finish KPI (fn=0.20 mm/rev, Re=0.80 mm, L=150 mm, 1,200 RPM) | Formula | Verified Output |
|---|---|---|
| Peak-to-Valley Roughness Rz (um) | ((fn_mm_rev ^ 2) / (8 * Re_mm)) * 1000 | 6.25 um (4.17 um @ Re = 1.20 mm) |
| Arithmetic Avg Roughness Ra (um) | ((fn ^ 2) / (18 * SQRT(3) * Re)) * 1000 | 1.60 um (1.07 um @ Re = 1.20 mm!) |
| Finish Pass Cutting Time (sec) | (Length_mm / (fn * Spindle_RPM)) * 60 | 37.50 s (unchanged @ Re = 1.20 mm!) |
| Bearing Journal Ra <= 1.65 Check | Ra_um <= 1.65 um | MEETS Ra <= 1.65 um SPEC |
Exact Parabolic Cusp Ra & Peak-to-Valley Rz Turning Roughness Formula
Validate 0 < f_n <= 0.8 * R_e (feed per rev cannot approach or exceed nose diameter without threading!) and positive Cut_Length and Spindle_RPM!
=LET(fn_r, B2, re_m, B3, ln_c, B4, rp_s, B5, IF(OR(fn_r <= 0, re_m <= 0, fn_r > 0.8 * re_m, ln_c <= 0, rp_s <= 0), "Invalid Turning Input", LET(ra_u, ((fn_r ^ 2) / (18 * SQRT(3) * re_m)) * 1000, IF(ra_u <= 1.65, "MEETS Ra <= 1.65 um SPEC", "TOO ROUGH FOR BEARING SEAT"))))Zero-Cycle-Penalty 1.20 mm Insert Nose Upgrade Check (Set Nose Radius R_e = 1.20 mm)
Swap the 0.80 mm finishing insert for a 1.20 mm nose radius (B3 = 1.20) at the exact same 0.20 mm/rev feed: Cycle Time E4 stays locked at 37.50 s while R_z E2 drops from 6.25 um to 4.17 um and R_a E3 improves from 1.60 um down to 1.07 um!
=IF(ISTEXT(E2), E2, ROUND(((B2 ^ 2) / (18 * SQRT(3) * B3)) * 1000, 2))Try the example
Use the original values or change them to test your understanding.