Lathe Set-Up: M20 × 2.5 Thread and a 1:50 Taper on the Same Machine
A 1.5 kW centre lathe has a 6 mm pitch lead screw and a change-gear set containing 20, 24, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 90, 100, 110 and 120 teeth. The job is an M20 × 2.5 external thread, 60 mm long, cut on a 45 mm diameter mild-steel bar held in a chuck. The thread is to be checked with a standard metric screw-thread micrometer and a go/no-go ring gauge. Later the same lathe must turn a 1:50 taper over a 100 mm length on a 40 mm diameter shaft mounted between centres, the full shaft being 200 mm long. The machinist proposes using the tailstock set-over method for the taper and a compound-rest swivel of 30° for the thread infeed; the cross-slide has 0.02 mm graduations.
1. Calculate the change-gear ratio for cutting M20 × 2.5 on the 6 mm lead screw. Select gears from the list and show the check calculation. What do you do if the 25/60 pair will not mesh at the available centre distance?
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Ratio = pitch to be cut / lead-screw pitch = 2.5/6 = 0.41667 = 5/12 = 25/60. Driver = 25 teeth on the spindle; driven = 60 teeth on the lead screw. Check: pitch cut = lead screw × driver/driven = 6 × 25/60 = 2.5 mm. Correct. If the 25/60 pair cannot be mounted because the centre distance is unsuitable, insert an idler of any convenient size, for example 40 T. An idler changes the direction of rotation but not the ratio: 6 × 25/40 × 40/60 = 2.5 mm. Set the idler in the quadrant and check for 0.1–0.15 mm backlash. Judgement: do not 'correct' the ratio with 24/60 = 0.4, which would cut 2.4 mm pitch and scrap the thread. If the lathe has a quick-change gearbox, set 2.5 mm and verify with a chalk mark and one spindle revolution before the first cut.
2. Calculate the thread depth and core (minor) diameter for M20 × 2.5 using the standard 0.6134P depth. If the compound rest is swivelled to 30°, what compound infeed corresponds to full thread depth, and how many 0.02 mm graduations is that?
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Depth per side h = 0.6134P = 0.6134 × 2.5 = 1.534 mm. Core diameter = d − 2h = 20 − 3.068 = 16.932 mm; equivalently d − 1.2269P = 20 − 3.067 = 16.933 mm. The two agree, and this is consistent with the standard external minor diameter for M20 × 2.5. Compound at 30°: depth is radial, but the compound feeds along its slide, so compound travel = h/cos 30° = 1.534/0.8660 = 1.771 mm. Graduations = 1.771/0.02 = 88.5, so set about 89 divisions from zero, allowing for the first scratch pass. Judgement: with the compound at 30° the tool cuts mainly on one flank, reducing tearing and improving finish; use the cross-slide for the final 0.05–0.10 mm to clean both flanks. Measure across the flanks with the thread micrometer, not over the crest with a vernier, and confirm with the ring gauge: the go gauge must screw on by hand and the no-go must not enter more than about two turns.
3. Calculate the small-end diameter and the tailstock set-over for the 1:50 taper. State whether the set-over method is suitable here, and give the compound-rest swivel angle as an alternative.
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1:50 means 1 mm diameter reduction per 50 mm of length. Over 100 mm: reduction = 100/50 = 2.000 mm, so small diameter d = 40 − 2 = 38.000 mm. Tailstock set-over = L × (D − d)/(2 × l), where L = 200 mm between centres and l = 100 mm taper length. Offset = 200 × 2/(2 × 100) = 400/200 = 2.000 mm. Half taper angle = tan⁻¹[(D − d)/(2l)] = tan⁻¹(2/200) = tan⁻¹ 0.01 = 0.573°, about 0°34′. The set-over method is suitable only because the shaft is between centres; it cannot be used with the job held in a chuck. Check that 2 mm is within the tailstock adjustment range, that the centre holes are deep enough to avoid binding, and that the centres are lubricated. If the compound rest is used instead, swivel it to 0.573°, not to a 1:50 mark. The limitation is travel: a compound slide typically has 75–100 mm travel and the taper is 100 mm long, so confirm the slide will not run out of travel; otherwise use a taper-turning attachment or the set-over method.