Mechanical Movement 10 — Curved Conoidal Pulleys
Why are some cone pulleys curved instead of straight?
Mechanism No. 10 from 507 Mechanical Movements: cone pulleys whose sides are slightly curved (conoids). With straight cones (No. 9) the sum D₁ + D₂ is constant, but an open belt needs an extra (D₁ − D₂)²/(4C) of length when the diameters differ, so a belt fitted in the middle is over-stretched at the ends. Shaping one cone so that π(D₁+D₂)/2 + (D₁−D₂)²/(4C) stays constant keeps the belt length, and therefore its tension, the same at every position. Simulation: straight driving cone 80 → 200 mm, curved driven cone, belt EA = 60 kN.
How it works, step by step
1. Straight cones (No. 9)
D₁ + D₂ is constant, so the ratio varies smoothly. Fit the belt with the fork in the middle…
2. The ends pull tight
…then slide it toward an end: an open belt needs 1.8 mm more length, so with straight cones tension would rise to 388 N.
3. Curve the cone
The driven cone is made 0.4 mm smaller here, exactly enough to keep the belt length unchanged.
4. Constant tension
With the curved profile the belt stays at 300 N everywhere, so it neither slips in the middle nor overloads at the ends.
The key idea
L(belt) = constant — Curving the cones keeps the belt length constant at every position, so tension stays even.
Conoids correct the (D₁−D₂)²/4C term of open-belt length, so the belt keeps one tension at every ratio.
Model and assumptions
The correction is small (about 5 mm on diameter at the ends for C = 400 mm) but the belt is stiff: without it, tension would swing from 300 N to over 700 N across the traverse. In the drawing the curvature is exaggerated 6× so it can be seen.
Frequently asked questions
Why curve the cones?
With straight cones an open belt needs more length near the ends; curving the profile keeps the required length constant.
What happens with straight cones?
Belt tension rises or falls as the belt moves, risking slip or overstretching.
Is the speed ratio still set by diameters?
Yes, n₂ = n₁ × D₁ / D₂ at the belt’s position.
Part of the 507 Mechanical Movements series
This is Movement No. 10 from Henry T. Brown’s 507 Mechanical Movements (1868), redrawn and simulated by BCRUZ.
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