Taper Turned With the Compound Rest
A lathe operator must turn a taper on a mild-steel shaft: large diameter 30 mm, small diameter 24 mm, over a length of 60 mm. The lathe has a compound (top) slide swivel graduated in degrees and a tailstock offset facility. The operator sets the compound to 5° by the graduated scale, takes a light trial cut, and measures the taper with a micrometer at each end: 30.02 mm and 27.10 mm. The finish is torn and the tool chatters near the small end. The workpiece is held in a three-jaw chuck with about 90 mm overhanging and is not supported at the free end.
1. Calculate the correct compound-rest angle and explain why the trial cut did not match the setting.
Read the worked answer
Half-angle from the axis:
- Radial reduction per side = (30 − 24)/2 = 3 mm
- tan θ = 3 / 60 = 0.05, so θ = 2.86°, roughly 2°52′.
The operator set 5°, which is nearly double the required angle, so the trial cut removed far too much at the small end - consistent with 27.10 mm instead of 24 mm.
The graduated scale on a compound rest is a convenience, not a precision reference. Scale markings are often coarse (1° divisions with no vernier), the zero may not be true, and the swivel can shift when clamped. A trial cut and measurement is the correct method, but the angle must then be corrected from the measured result, not simply re-read from the scale. Using the measured diameters, the achieved half-angle was tan⁻¹((30.02 − 27.10)/2 / 60) ≈ 1.39°, so the slide must be rotated by roughly the difference between that and 2.86°, then re-trial-cut and re-measured until the taper is within the drawing tolerance.
2. Diagnose the torn finish and chatter, and state how the setup should be changed.
Read the worked answer
The 90 mm overhang in a three-jaw chuck with no tailstock support is the primary cause. At the small end the bar is least stiff, so tool pressure deflects it, producing chatter and a torn surface. Deflection also means the measured diameters are unreliable - the bar springs back after the tool passes.
Changes:
- Support the free end with a tailstock centre (face the end and drill a centre hole first) or use a fixed steady if the work must pass through the spindle. This alone will remove most of the chatter.
- Reduce overhang to the minimum the taper allows.
- For the finish pass, use a sharp, correctly ground tool with a small nose radius, positive rake suited to mild steel, a fine feed and a moderate depth of cut. Too light a finishing cut can rub and tear rather than cut.
- Apply cutting fluid and check that spindle speed suits the diameter; the small end runs at lower surface speed but is the least rigid point.
- Re-measure only after the work is supported and the machine has settled, since the earlier readings were taken on a deflecting bar.