Lathe: Morse tapers, taper turning and knurling

November 3, 2026  ·  Indumor

The chuck hangs crooked off the drill press spindle. Somebody dropped it yesterday, rammed it back in, and now the 1/2 in drill wobbles. The machinist knocks it out with the drift, runs a finger down the shank and feels the flattened chip at once. A fine stone, a rag, and the taper grabs again with a sharp slap of the palm. That is the whole physics of the Morse taper: two clean conical surfaces that hold by friction alone.

Why a Morse taper holds by itself

The half-angle of a Morse taper runs between 1°26′ and 1°30′. With an angle that small, the seating blow multiplies against the socket walls and friction does the rest: it is a self-holding taper. The tang does not carry torque; it is there so the drift can push the shank out. If the tang is doing the work, the taper is slipping, and a slipping taper chews up the spindle socket.

The seven sizes do not share one taper. Each has its own, from 1:19.002 (No. 5) to 1:20.047 (No. 1), because the series was laid out in inches per foot in the 1860s and every size was rounded on its own. The number is read off in seconds with a caliper: D is the mouth of the socket, 0.475 in (12.065 mm) for MT1, 0.700 in (17.78 mm) for MT2, 0.938 in (23.825 mm) for MT3; on a shank the large end measures a little more (12.2, 18 and 24.1 mm).

Ratio, D, l1 and b are those of ISO 296 / DIN 228-1. L, d and T are the inch values of ASME B5.10 (plug depth 2, 2 1/8, 2 9/16, 3 3/16, 4 1/16, 5 3/16 and 7 1/4 in); ISO 296 gives L, in the same order, as 50 / 53.5 / 64 / 81 / 102.5 / 129.5 / 182 mm, with d 6.4 / 9.4 / 14.6 / 19.8 / 25.9 / 37.6 / 53.9 mm. Form A of DIN 228-1 replaces the tang of form B with the drawbar thread in the last column.

A small lathe tailstock takes MT2 or MT3 and the spindle MT4 or MT5. Milling arbors use form A, with a drawbar thread instead of a tang, because interrupted cuts loosen any taper that is not pulled from behind.

Metric, 7/24, Jacobs and B tapers

Above MT6 the ISO 296 metric series carries on at 1:20, named by the large diameter, 80 to 200. Below MT0 sit the metric 4 and 6, for very small tools. At a half-angle of 1°25′56″ they are still self-holding.

Milling-machine tapers are a different animal. The 7/24 steep taper (ISO 297) opens 16°35′ and will not hold itself: it locates the holder, but torque goes through the drive keys and retention comes from a drawbar or a pull stud. D1 and the taper are the same in every variant; the flange and the stud are not, and that is where imported holders bite.

These are the usual threads; before ordering a pull stud, confirm the thread and form in the machine manual.

Drill chucks use two families. Jacobs tapers are American and every number has its own taper, 0.59 to 0.98 in per foot; JT33 and JT6 go on most 1/2 in chucks. The B tapers of DIN 238 are the metric alternative, short lengths of a Morse taper with the angle of MT1, MT2 or MT3; B16 is the common one on 13 and 16 mm chucks. Filing one to fit the other ends with a chuck that wobbles.

Taper ratio, angle and taper per foot

Taper is the difference in diameters divided by the length over which it happens. It is written as a ratio (1:20 means the diameter drops 1 mm every 20 mm of length), in mm per meter, as a percentage, or as taper per foot on American drawings. A protractor reads the included angle α; the compound rest is set to half of it, α/2, and that mix-up has ruined more tapers than anything else. On a drawing (ISO 3040) you give the ratio or the angle; one of the two is enough.

For 1:k the half-angle is the arctangent of 1/(2·k): 1°25′56″ for 1:20, 2°51′45″ for 1:10. Tangents are in chapter 23; tapered pipe threads, which are 1:16 tapers (3/4 in per foot), in chapter 6.

Turning a taper: tailstock offset or compound rest

There are three ways to cut a taper on an engine lathe. For short tapers of any angle (a Morse shank, a valve seat) swing the compound rest to the half-angle and feed by hand; its travel, 3 to 5 in on most lathes, sets the length limit. For long, shallow tapers between centers, set the tailstock over: the carriage runs under power feed and the work, spinning on a tilted axis, comes out tapered. The third is the taper attachment, if the lathe has one.

The compound scale reads in whole degrees and tapers want minutes. The trick is a dial indicator against a known good shank: run the compound 100 mm and the needle must travel what the arithmetic says, 2.50 mm for an MT2 (0.025 in per inch of travel). Check the finished taper with three stripes of Prussian blue along the shank and a quarter turn in the socket: blue wiped off at the mouth means the angle is too big; wiped off at the bottom, too small.

Tool angles

A cutting tool is a wedge, and the wedge is defined by three angles that add up to 90°. The clearance (relief) angle α keeps the flank off the machined surface; the wedge angle β is the body of the tool; the rake angle γ tilts the face the chip slides on. A large rake cuts with less force in soft materials; a small or negative rake survives interrupted cuts and hard materials.

The table gives values for hand-grinding HSS tool bits, and they are guide values: they depend on the rigidity of the setup, the feed and whether the cut is continuous. Carbide inserts come with their geometry pressed in and usually run with less rake. Speeds and feeds by material are in chapter 9.

Knurling to DIN 82

Knurling does not cut, it displaces. The wheels press in and the metal flows, half into the groove and half upward, so the part ends up larger than it started. DIN 82 names the patterns with three letters and fixes the pitches t from 0.5 to 1.6 mm; it does not tie them to the diameter, though in practice they grow with it; American knurls are specified instead by teeth per inch (TPI), so a 1 mm pitch is close to 25 TPI.

Two sums before you knurl. The blank is turned under the finished size (half a pitch under for straight and diagonal patterns, two thirds of a pitch for raised-point diamonds, one third for depressed-point ones), because the metal that rises makes up the difference. And the circumference of that blank has to be a whole multiple of the pitch; if it is not, on the second turn the wheel lands between two grooves of the first and the pattern doubles.

Blank diameter d2 = finished size d1 minus the undersize; it is a reference value, since the formula does not allow for the material. For the pattern to close without double tracking, d2 should be a multiple of t/π (with t = 1 mm: 19.74 or 20.05 instead of 20).

The rest is practice. Low speed, plenty of oil, the wheels on center and a firm first pass, because the pattern is set in the first revolution. If it looks doubled on the second, do not lean on it harder: back off, clean up and start again at another spot. For thin tubing or a long shaft that bows, cut knurling, with skewed wheels that actually remove a chip, does the job without the pressure.

Frequently asked questions

How do I tell which Morse taper my drill press or lathe has?

Measure the mouth of the socket with a caliper: 12.065 mm (0.475 in) is MT1, 17.78 mm (0.700 in) is MT2, 23.825 mm (0.938 in) is MT3, 31.267 mm (1.231 in) is MT4 and 44.399 mm (1.748 in) is MT5. On a shank the large end measures a little more: 12.2, 18, 24.1, 31.6 and 44.7 mm. The taper length confirms it: 65 mm for MT2, 81 mm for MT3.

How far do I swing the compound rest to turn a taper?

To the half-angle α/2, from tan(α/2) = (D − d)/(2·L). A 1:10 taper takes 2°51′45″; a Morse taper, between 1°26′ and 1°30′. Since the compound scale reads in whole degrees, the fine setting is done with a dial indicator on a known shank.

What is the difference between a B16 taper and a Morse taper?

The B16 (DIN 238) is a short length of the MT2 taper: same angle, but 24 mm long and 15.733 mm at the large end. It goes on 13 and 16 mm drill chucks. It is not interchangeable with Jacobs tapers, which have different angles.

Sources: DIN 228-1:1987 (Morse and metric tapers, taper shanks); ISO 296:1991; ISO 297 (7/24 tapers); DIN 69871; ISO 7388-1; MAS 403 (BT); ASME B5.50 (CAT); DIN 2080; DIN 238 (B tapers for drill chucks); ASME B5.10-1994, Morse, metric and Jacobs tapers; ISO 1119 and DIN 254 (standard tapers); ISO 3040 (dimensioning of cones); DIN 82 (knurls); ISO 3002-1 (tool geometry); guide values for tool angles from shop handbooks.