Corner Arc and Pitch-Circle Set-Out on a Link Plate
A small fabrication shop is producing a 180 mm × 120 mm link plate from 6 mm mild steel. The profile has two straight edges meeting at 50° at vertex A, and a 35 mm radius corner arc that must blend tangentially into both edges. Six Ø10 holes are required on a 70 mm pitch circle centred on the plate; the first hole lies on a line 15° above the horizontal centreline and the rest are equally spaced. The trainee set out the arc by marking a centre 'about 80 mm from A' and drawing until it looked right, and stepped the holes round the pitch circle with dividers set once by trial. The first article shows a 0.3 mm step where the arc meets each straight edge, and the sixth hole is 0.8 mm out of position, so the mating cover will not seat. The CNC operator insists the machine holds 0.05 mm and that the drawing is at fault.
1. State where the arc centre must lie, calculate its position and the two tangent points, and explain why the trainee's method produced a visible step.
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Where the centre lies. A circle tangent to two intersecting lines has its centre on the angle bisector, at a perpendicular distance R from each line. So:
- Offset each edge inward by 35 mm; the intersection of the two offsets is the centre O. Equivalently, bisect the 50° angle and step off
- AO = R / sin(θ/2) = 35 / sin 25° = 35 / 0.4226 = 82.80 mm.
The trainee's 'about 80 mm' was 2.8 mm short and, worse, was measured without reference to the bisector, so O was not equidistant from the two edges.
Tangent points. Drop perpendiculars from O onto each edge; the feet T1 and T2 are the tangent points. AT = R / tan(θ/2) = 35 / tan 25° = 35 / 0.4663 = 75.06 mm along each edge.
Why the step appears. A line and an arc meet smoothly only when the radius at the junction is perpendicular to the line. With an eye-placed centre the perpendicular distance to one edge was slightly under 35 mm and to the other slightly over, so the arc met each edge at a small angle - a kink, not a tangency. A 0.3 mm step is consistent with a centre error of only 0.3–0.4 mm in perpendicular distance, which is invisible when setting out by eye but obvious once the profile is cut.
Check before release. Measure the perpendicular distance from O to both edges (both must be 35 mm) and check the tangent-point chord T1T2 = 2R cos(θ/2) = 2 × 35 × cos 25° = 63.4 mm.
2. The six holes are equally spaced on a 70 mm pitch circle. Calculate the chordal spacing and the coordinates of the first two holes relative to the plate centre, then explain how the holes should be set out so the sixth cannot drift.
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Chord. For n holes the chord is 2R sin(180°/n) = 2 × 35 × sin 30° = 35.000 mm. For a hexagon the chord equals the radius exactly - a useful check, because a divider setting that does not agree with the 35 mm radius is wrong before any stepping starts.
Coordinates (origin at plate centre, x to the right, y up):
- Hole 1 at 15°: (35 cos 15°, 35 sin 15°) = (33.81, 9.06)
- Hole 2 at 75°: (35 cos 75°, 35 sin 75°) = (9.06, 33.81)
- remaining holes at 135°, 195°, 255°, 315°.
Why the sixth hole drifted 0.8 mm. Stepping a divider six times round a circle accumulates every setting error and every punch mark error; the last hole carries the sum of all five previous steps. Setting the dividers by trial against the arc, rather than to the calculated chord, adds a systematic error at every step, and the closing error is never distributed.
Correct set-out. Draw the Ø70 pitch circle and the 15° centreline, strike R35 from the centre to fix hole 1, then locate each remaining hole independently - by coordinates from the centre, by a CAD polar array, or by 60° set-square construction - rather than by stepping. After set-out, check the across-centre diameters 1–4, 2–5 and 3–6; all three must pass through the plate centre. A drift shows immediately as three chords that do not intersect at one point.
3. Recommend the changes to the drawing and the inspection routine that would stop this recurring.
Read the worked answer
Drawing changes
- Annotate the arc 'R35 TANGENT TO BOTH EDGES' and dimension the 50° included angle plus one locating dimension to the arc centre (e.g. 82.8 mm along the bisector), so the centre is defined, not estimated.
- Replace the stepped hole set-out with a single note: '6 HOLES Ø10 EQ SP ON Ø70 PCD, FIRST HOLE 15° ABOVE HORIZONTAL CENTRELINE, POSITION FROM CENTRE DATUM' and, for CNC or CMM, add an X–Y coordinate table.
- Give the pitch circle a positional tolerance (e.g. Ø0.2) rather than leaving position untoleranced, and identify the plate centre as datum B and an edge as datum C so the table means one thing only.
Inspection routine
- Plot or check the profile at 1:1 before issue. A tangency fault of 0.3 mm is invisible on a 1:2 plot; at 1:1 a straight edge laid along the flat will show light at the junction.
- Verify the first article on a CMM or optical comparator against the coordinate table, not by calipers across edges.
- Have the setter confirm both perpendicular distances to the arc centre with a height gauge before pressing 'cycle start'.
The machine was not at fault: the geometry on the drawing was under-defined, so the drawing did not control the part.