Reverse-curve link: choosing a buildable tangent radius
A steel tension link joins two pivots whose centres O1 and O2 lie 80 mm apart on a horizontal line. The left boss has radius 12 mm, the right boss radius 7 mm. The outline is closed by a reverse curve: one arc of radius R is externally tangent to the left boss and internally tangent to the right boss, running above the centre line, while a mirror-image arc closes the lower side. The draughtsman proposes R = 30 mm to keep the link narrow, but the workshop foreman prefers R = 40 mm and asks whether the geometry is buildable before the CNC program is generated. Both boss circles must lie wholly inside the outline, and every junction must be tangent so that no undercut or kink appears at the transition. The centre distance and boss radii are fixed; only R may change, and the link must remain symmetric about the horizontal centre line.
1. Test whether R = 30 mm and R = 40 mm are geometrically feasible. Show the governing distances and the triangle condition.
Read the worked answer
Let C be the centre of the connecting arc. For external tangency to the left boss, the centre distance is the sum of radii:
CO1 = R + 12.
For internal tangency to the right boss, the connecting arc encloses the boss, so
CO2 = R − 7, which also requires R > 7.
With O1O2 = 80 mm, the two locus circles must intersect. A necessary and sufficient condition is
|CO1 − CO2| ≤ 80 ≤ CO1 + CO2.
For R = 30: CO1 = 42 mm, CO2 = 23 mm; sum = 65 mm < 80 mm. The circles are separate, so no centre C exists. R = 30 is not buildable.
For R = 40: CO1 = 52 mm, CO2 = 33 mm; sum = 85 mm ≥ 80 mm, and difference = 19 mm ≤ 80 mm. Two intersections exist, one above and one below the centre line, so R = 40 is feasible.
2. Describe the construction of the upper arc for R = 40 mm and state exactly where the two tangent points lie. Explain why the tangency conditions make the junctions smooth.
Read the worked answer
Construction: draw O1O2 = 80 mm. With centre O1 and radius 52 mm draw an arc; with centre O2 and radius 33 mm draw a second arc. Their intersections are the two possible centres C and C′. Choose the upper intersection C for the upper profile.
Tangent point T1: because the connecting circle and the left boss are externally tangent, T1 lies on segment O1C with O1T1 = 12 mm and CT1 = 40 mm. Equivalently, CT1 = CO1 − 12 = 52 − 12 = 40 mm.
Tangent point T2: because the connecting circle encloses the right boss, T2 lies on the ray from C through O2, beyond O2, with O2T2 = 7 mm and CT2 = 40 mm. Check: CO2 = 33 mm and 33 + 7 = 40 mm.
Draw the arc of radius 40 mm centred at C from T1 to T2. At each tangent point the two curves share a common normal through their centres, so the tangent directions coincide; there is no kink. Mirror the construction about O1O2 for the lower profile.
3. The two intersections of the centre-locus circles are both mathematically valid. How should the draughtsman use them, and what practical checks should be made before accepting the R = 40 mm outline?
Read the worked answer
The two intersections are symmetric about O1O2. The upper intersection C generates the upper connecting arc; the lower intersection generates the lower arc. They should not both be used for the same half of the outline. Use C for the upper profile and its mirror for the lower profile, then close the outline with the outer arcs of the two bosses.
Practical checks:
- Confirm that each boss lies on the correct side of its tangent point: the left boss is outside the connecting circle and touches externally; the right boss is inside and touches internally.
- Check that the connecting arc does not cross either boss elsewhere. Tangency guarantees local contact, but the full arc must be trimmed at T1 and T2; continuing it would intersect the boss circles.
- Confirm that the upper and lower arcs plus the boss arcs form one closed non-self-intersecting profile.
- Reject any solution in which R ≤ 7 mm, because internal tangency to the 7 mm boss is then impossible.
R = 40 mm is therefore the acceptable choice; R = 30 mm must be abandoned unless the centre distance or boss radii change.