Link Plate Profile: A Blending Arc That Cannot Be Built
A link plate is being set out at 1:1 on A3 for a CNC router. Two boss circles of Ø40 mm and Ø24 mm have centres 70 mm apart on a horizontal centre line. The upper profile is to be a single blending arc of radius 50 mm, concave upwards, joining both boss circles tangentially, with the plate material below the arc. A candidate has drawn construction arcs of radius 50 mm from each boss centre, found no intersection, and proposes to move the smaller boss 2 mm closer so that the construction will work. A straight tangent line is also needed from the Ø40 mm boss to a 20 mm wide tab on the left. The profile must be checked before the router program is generated, because a rejected plate wastes a 6 mm aluminium blank.
1. Use the construction distances to show whether the R50 arc is possible, and state the minimum radius that keeps the 70 mm centres.
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For an arc that contains both boss circles and is tangent to them from outside (so the profile is concave upwards with material below), the arc centre must be R − r from each boss centre, not R + r and not R.
- Distance to the Ø40 boss centre: 50 − 20 = 30 mm.
- Distance to the Ø24 boss centre: 50 − 12 = 38 mm.
- Largest possible separation of the arc centre from the two boss centres: 30 + 38 = 68 mm.
Since 68 mm < 70 mm, the two construction arcs cannot intersect and no centre exists. The R50 arc is geometrically impossible. The candidate's error is drawing construction arcs of radius R from the boss centres, which applies to a different tangency case.
Keeping the 70 mm centres, the condition is (R − 20) + (R − 12) ≥ 70, so 2R ≥ 102 and R ≥ 51 mm. Use R = 52 mm, not 51, since 51 gives a single grazing intersection and no usable arc.
2. Describe the construction for the corrected R52 arc, including how the tangent points are found.
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- With the Ø40 boss centre as centre, draw a construction arc of radius 52 − 20 = 32 mm.
- With the Ø24 boss centre as centre, draw a construction arc of radius 52 − 12 = 40 mm. These intersect at O, the centre of the blending arc.
- Draw a line from O through each boss centre and extend it to cut the boss circle on its far side. Those intersections are the tangent points T1 and T2, because circles that touch internally have their tangent point on the line of centres.
- With O as centre and radius 52 mm, draw the arc from T1 to T2. It should run smoothly into both boss circles with no corner.
- Mark O and the construction arcs lightly and leave them, since a set-out drawing must show how the profile was obtained. Check by eye along the profile: any kink means a tangent point or radius is wrong.
3. Describe how to construct the straight line, tangent to the Ø40 mm boss, that leads to the 20 mm wide tab on the left.
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Treat the tangent line as a line at distance 20 mm (the boss radius) from the boss centre.
- Draw the tab edge line in its correct position from the tab dimensions.
- Offset that line by 20 mm towards the boss using a parallel line construction; the offset line passes through every point that is 20 mm from the tab edge and therefore can be tangent to the boss.
- Where the boss circle cuts the offset line, mark the tangent point. A perpendicular from the boss centre to the tangent line must meet the circle at exactly that point, which is the check.
- Draw the tangent from the tab corner to that point and trim the boss circle beyond it.
If the tangent point is not on the boss circle, the offset is the wrong distance or the tab edge is not parallel to the tangent line; do not force a line by eye, because the router will cut the error.