Surface Roughness (Ra, Rz) and Finish Milling: Theoretical Calculation, Stepover Selection and Measurement

Surface roughness is the single criterion that determines both part acceptance and cycle time in finish milling. This guide covers what parameters such as Ra and Rz measure, how theoretical roughness is calculated from the nose radius and the feed, how the target value is reached through stepover selection in ball nose milling, and which rules govern the measurement. Coating, material group and cutting speed selection are outside its scope.

1. What Ra, Rz and Rmax actually measure

Roughness parameters are calculated from a filtered profile. Once form and waviness have been separated from the measured raw profile, what remains is the roughness profile. The parameters are defined on this profile.

Ra on its own can be misleading. The averaging operation dissolves a single deep scratch among hundreds of measurement points. Where a sealing surface, a region carrying fatigue loads or a plain bearing surface is concerned, Rz or Rmax has to be specified alongside Ra.

There is no reliable conversion factor between Ra and Rz. On milled surfaces the ratio mostly varies between three and seven; as the shape of the profile changes, so does the ratio. For this reason, whichever parameter is written on the drawing is the one that has to be measured.

Direction matters: Roughness is directional. On a milled surface, a measurement taken perpendicular to the feed direction captures the feed marks; taken along the feed direction it captures the stepover scallops. A difference of several times between the two measurements is normal.

2. The ISO 21920 series and the status of the former standards

Published in December 2021, the ISO 21920 series brought the surface texture standards based on the profile method together under a single roof. The series consists of three parts: ISO 21920-1 covers the indication on the drawing, ISO 21920-2 the parameter definitions, and ISO 21920-3 the verification and decision rules. With this publication ISO 1302, ISO 4287, ISO 4288 as well as ISO 13565-2 and ISO 13565-3 were withdrawn.

The changes that stand out in practice are the following:

• Change of terminology. The term "section length" is used in place of "sampling length". The name of Rz has been changed from "mean roughness depth" to "maximum height". The former Rmax equivalent has been defined as Rzx.
• Basis of calculation. Height parameters such as Ra and Rq are now calculated as a single value directly over the evaluation length. Peak-to-valley parameters continue to be calculated over section lengths for stability.
• Defaults. ISO 21920-3 defines setting classes in order to reduce the choices left to the personal interpretation of the inspector. The default tolerance rule is the maximum rule; for the 16 percent rule of former practice to apply, it has to be stated explicitly on the drawing.
• Order of operations. The sequence of the filtering and form removal operations has been changed.

Since the definitions of Ra, Rq, Rt and Rsk are preserved in essence, existing drawings do not become invalid. On the other hand, measuring the same surface according to the old and the new rules can produce different numbers, particularly on profiles that contain form. Where areal evaluation is required, the ISO 25178 series is used.

3. Theoretical surface roughness: nose radius and feed

A cutter with a rounded nose leaves the trace of a circular arc on the surface at every feed increment. The material remaining between successive traces forms the geometric roughness of the surface. When vibration, wear and material behaviour are left out of account, roughness depends on two quantities alone: the feed per revolution f and the nose radius rε.

The commonly used approximations are the following:

• Total height: Rt ≈ f² / (8 · rε)
• Average roughness: Ra ≈ f² / (32 · rε)

In both formulae f and rε are in millimetres; the result comes out in millimetres and is converted to micrometres by multiplying by 1,000. The relationship is not linear: when the feed is halved the theoretical Ra falls to a quarter, and when the nose radius is doubled it falls by half.

The table below gives the theoretical Ra values calculated from the formula, in micrometres.

The table shows only the geometric lower limit. The actual measurement always comes out above this value. In practice, keeping the theoretical value at around half of the target tolerance and leaving the remaining margin to the effects of vibration and wear is an established approach.

The formula has one limit: when the feed per tooth drops below the minimum chip thickness, the cutting edge ploughs and rubs the material instead of cutting it. In this region, reducing the feed further does not improve the surface but degrades it and shortens tool life. How parameter selection is built up is dealt with separately under cutting speed, spindle speed and feed calculation.

4. Scallop height in ball nose milling

When a three-dimensional surface is machined with a ball nose cutter, crescent-section scallops remain between neighbouring tool paths. The height of these scallops is defined by the tool radius R and the stepover width ae:

h = R − √(R² − (ae/2)²)

When the stepover remains small relative to the tool diameter, this expression reduces to the approximation h ≈ ae² / (8 · R). The approximation is accurate enough for shop floor calculations. The remaining scallop height corresponds to the peak-to-valley height of the profile measured perpendicular to the feed; owing to the same circular arc geometry, the Ra value stays roughly of the order of a quarter of this height.

The table below gives the scallop height for different tool diameters, in micrometres.

Since the relationship is quadratic, doubling the stepover quadruples the scallop. For the same reason, reducing the stepover by about 30 percent is enough to halve the remaining scallop; halving the stepover reduces the scallop to a quarter and doubles the machining time.

Stepover selection for a target roughness

Working backwards from the target Ra value, the largest usable stepover is found from the relationship ae ≈ √(32 · R · Ra). The table below gives the results of this calculation, in millimetres.

The values in the table are the geometric upper limit and applying them directly is not recommended; under real conditions roughly half of these values is selected, leaving a margin for vibration and wear. The table also shows why a large diameter is advantageous: a 16 mm ball nose cutter allows a stepover about 60 percent wider than a 6 mm one at the same roughness.

The second determining factor in ball nose milling is the centre region of the tool. Since the cutting speed drops to zero at the centre, this region has to be kept out of the cut. Sandvik Coromant recommends tilting the spindle or the workpiece by 10–15°; this moves the cut away from the centre, raises the effective cutting speed and improves the surface markedly. The same source recommends taking the feed per tooth roughly equal to the stepover width in order to obtain a symmetrical texture in finishing. Ball nose cutter geometries can be examined in the ball nose end mills category.

5. Four factors that break the theory

The difference between the measured roughness and the calculated value comes from the following four mechanisms.

Chatter. Self-excited vibration leaves regularly spaced wave marks on the surface. These marks alone can increase Ra several times over and are usually recognised from the sound of the cut. Reducing the overhang-to-diameter ratio, changing the width of cut and moving into a stable region by means of a spindle speed sweep are the principal remedies; the subject is treated in detail under chatter in milling: causes and solutions.

Runout. Radial runout causes the teeth to take different amounts of chip. As runout grows, the load concentrates on a single tooth and what appears on the surface is not the number of teeth but the feed mark of a single tooth. In that case the actual feed per tooth is several times larger than the calculated one and the theoretical roughness calculation becomes invalid. In precision finishing, runout has to be kept as small as possible; the effect of the holder type on runout is compared in the tool holders guide.

Tool wear. As flank wear progresses, the effective radius of the cutting edge grows, the edge dulls and the amount of rubbing against the surface increases. The roughness measured in die and mould finishing degrades gradually as the wear limit is approached. For this reason the finishing tool has to be kept separate from the roughing tool, and the life of the finishing tool has to be limited according to roughness measurement. Wear mechanisms are examined under tool wear mechanisms and tool life.

Chip re-cutting. Chips that cannot be evacuated from the cutting zone become trapped between the cutting edge and the machined surface and are embedded in the surface or leave scratches. In adhesive materials such as aluminium and austenitic stainless steel, this effect combines with the formation of a built-up edge on the cutting edge. At low cutting speeds built-up edge formation increases, it breaks away periodically and sticks to the surface, and degrades the roughness markedly. Raising the cutting speed above the built-up edge region and evacuating the chips with pressurised coolant or air produce a measurable improvement on the finished surface.

6. The effect of climb and conventional milling on the surface

The difference between the two methods lies in how the chip thickness varies.

Sandvik Coromant states that climb milling should always be preferred as long as the machine, the clamping and the workpiece permit it. Conventional milling makes sense only on older machines with backlash in the feed screw, on heavily scaled cast and forged surfaces, or in cases where the cutting edge is not wanted to dive directly into the hard skin.

In practice, planning the finishing pass one-directionally (climb milling in one direction, in the air on the return) eliminates the texture difference between the outward and return passes. Cycle time increases somewhat; in return, no difference in marking arises on the surface due to the change of direction.

7. Measurement: profilometer, cut-off and comparison specimen

The most widespread method is the stylus profilometer. A diamond-tipped stylus is drawn along the surface at constant speed and the vertical movement is digitised. Stylus geometry, measuring force and digitisation conditions are defined within ISO 3274. The radius of the stylus tip limits the width of the narrowest valley that can be measured; on very fine surfaces a small tip radius is required.

The most critical setting of the measurement is the cut-off (λc) value. The cut-off is the limiting wavelength of the filter separating roughness from waviness, and it also determines the section length. The evaluation length is by default five section lengths. The selection is made according to the expected Ra or Rz value; the table given in ISO 4288 for aperiodic profiles is below.

Most milled parts fall into the band of 0.8 mm cut-off and 4 mm evaluation length. If the cut-off is chosen smaller than required, machine-induced waviness is filtered out and left outside, and the surface looks better than it is. If it is chosen larger than required, waviness is mixed into the roughness and the result comes out worse than it is. For this reason the cut-off value also has to be stated in the measurement report; an Ra value without a stated cut-off is not comparable.

The second method is the comparison specimen. ISO 2632-1 covers turned, ground, drilled, milled and planed surfaces; ISO 2632-2 covers spark-eroded, blasted and polished surfaces. The specimens are compared visually and by touch with surfaces produced by a similar method and having a similar lay direction. They are convenient for rapid pre-screening at the machine; they are not sufficient for numerical tolerance verification.

Source of measurement error: When the direction and position of the measurement are not written in the report, the result is not repeatable. On milled surfaces the measurement has to be taken perpendicular to the feed marks and at several representative points on the surface, and it has to be repeated at the same points throughout the life of the finishing tool.

Sources

• ISO 21920-1:2021 — Geometrical product specifications (GPS) — Surface texture: Profile — Part 1: Indication of surface texture
• ISO 21920-2:2021 — Surface texture: Profile — Part 2: Terms, definitions and surface texture parameters
• ISO 21920-3:2021 — Surface texture: Profile — Part 3: Specification operators
• ISO 4288:1996 — Rules and procedures for the assessment of surface texture (withdrawn; referenced for the cut-off and evaluation length tables).
• ISO 3274:1996 — Nominal characteristics of contact (stylus) instruments.
• ISO 2632-1:1985 and ISO 2632-2:1985 — Roughness comparison specimens.
• Digital Surf — What are the differences between ISO 4287 and ISO 21920? digitalsurf.com
• Sandvik Coromant — Profile milling and Down milling vs. up milling. sandvik.coromant.com