Flexible Belts and Sheaves

§1Traction, not teeth

A belt transmits by friction and forgiveness — and half its virtues are things a gear train cannot do, starting with the licence to slip.

Where gears mesh, belts grip, and the difference writes the belt’s character. It spans long centres for the cost of a strip of rubber and cord, where gearing the same distance would take a train of expensive wheels; it runs quiet and smooth, its compliance filtering the torsional chatter that gears pass straight through; it forgives modest misalignment and sloppy centres that would destroy a mesh; and — the property with a fuse’s value — it can slip: a jammed machine stalls its belt and squeals, spending rubber instead of shafts, the softest overload device in the previous page’s family. The prices are the same facts inverted. Friction drive means a percent-or-so of creep even in health (the tight side’s stretched belt contracting across the driven pulley), so the speed ratio is approximate — disqualifying belts from timing duty until §5’s toothed answer; capacity depends on tension the bearings must carry all day; and rubber ages, so a belt is a scheduled consumable. The engineering content of the page is then exactly two multiplications: how hard friction can grip (§2, §3), and what that grip is worth in power (§4).

§2The capstan law

Wrap a flexible strand on a drum and each element of arc lets tension grow in proportion to itself — the compound-interest condition, whose solution is the exponential every sailor exploits.

The mechanism deserves one clear sentence: at every element of wrapped arc, the strand’s tension presses it onto the drum, that pressure buys friction, and the friction lets tension step up across the element — so tension compounds along the arc, and compounding along a length is an exponential in that length. Run the numbers for the hero’s drive: μ = 0.3 on the small pulley’s θ = 160° of wrap (2.793 rad) gives a limiting ratio of e^0.838 = 2.31 — the tight side can carry at most two-and-a-third times the slack side before the belt lets go and slides. Two working corollaries fall straight out. Wrap is currency: the exponent buys grip in degrees, the small pulley always has the least and therefore governs the drive, and an idler pressed on the slack side earns its keep precisely by adding θ where it is scarcest. And the law’s ceiling character matters: e^μθ is what friction can sustain; what it does sustain is set by how hard the belt is pre-tensioned — the lever §4 prices. For anything more, change μ itself — which is the vee’s trick, next.

§3The vee’s wedge

Fold the belt into a trapezoid and seat it in a matching groove, and the groove’s flanks squeeze it like the vee thread squeezed its nut — friction is multiplied by geometry before the exponent ever sees it.

The mechanism is the threads section’s 1/cos φ surcharge, welcomed instead of resented. A belt pressed radially into a groove of half-angle around 19° bears on the flanks with a normal force magnified by the wedge, so the effective friction becomes μ/sin 19° = 0.921 from a material μ of 0.3 — a multiplication of 3.07× sitting inside the exponent. The same 160° of wrap that limited the flat belt to 2.31 now sustains e^{0.921 × 2.793} = 13.1: nearly six times the grip ratio from the same rubber, the same wrap and the same shaft pull, which is the entire, sufficient case for the V-belt’s century of dominance in machine drives. One geometric law polices the gift: the belt must ride on the flanks with clearance at the groove bottom — the hero’s inset — because a belt worn narrow (or a groove worn wide) enough to bottom sits on a flat, loses the wedge, and reverts to flat-belt grip while looking perfectly installed: the classic slipping drive that “has a good belt in it”. Hence the sheave gauge of §6, and the small print behind multi-belt drives: several vees share load only if they share length, so belts are replaced in matched sets, never one at a time.

§4A drive, priced in kilowatts

Power is the tension difference times the belt speed — so the whole §2–§3 apparatus exists to let T2 be small, and one table prices the hero’s drive end to end.

§5Timing belts

Mould teeth onto the belt and pockets into the pulley, and the drive changes species: no slip, no creep, an exact ratio — a chain’s certainty at a belt’s weight and silence.

The synchronous (toothed) belt is positive drive in rubber: tension cords carry the load, moulded teeth index it, and the ratio is as exact as gearing — which is why camshafts, servo axes, printers and every phase-critical light drive migrated to it. Its economics are gentle in exactly the place §4 finished: needing no friction, it needs almost no pre-tension, so a timing drive loads its bearings a fraction of what an equivalent vee does, runs cooler, and wastes less. The bill is the fuse clause and the discipline clause. There is no slip: a jam now stalls the motor, shears the belt’s teeth, or breaks something dearer — the timing belt hands back the overload protection §1 counted as a belt virtue, and drives that jam by trade keep a limiter in the line. And positive engagement is fussy: alignment must be true and the belt tracked by flanged pulleys, debris in a tooth pocket is a ratcheting failure, and the cords (glass or aramid) hate the two abuses that kill most timing belts in the field — crowbarring the belt over a flange at fitting, and kinking it in storage — both of which crack cords invisibly and schedule a snap for later. Choose it where the ratio is the point; keep the vee where forgiveness is.

§6Sheaves and practice

Most belt failures are sheave failures wearing a belt’s clothes — the drive is maintained at the grooves, the alignment and the fitting, and the belt merely reports the result.

The groove first: sheaves wear hollow and wide, and a dished flank lowers the belt until §3’s bottoming clause triggers; the check is a groove gauge held to the profile (daylight under the gauge condemns the sheave), and fitting a new belt into worn grooves is the standard way to buy a week of drive for the price of a belt. Alignment next: offset or angled sheaves roll and flip vees, wear one flank shiny, and walk timing belts into their flanges — a straightedge or laser across both rims, checked in both planes, is minutes of work with the whole shaft-alignment page (which closes this section) standing behind it. Diameter discipline: every trip around a small sheave bends the tension cords through their tightest arc, and bending is fatigue duty, so each belt section carries a minimum sheave diameter that is a life number, not a suggestion — the small pulley governs the drive twice over, in §2’s wrap and here in fatigue. And fitting manners close the loop: slacken the centres and walk the belt on by hand; the screwdriver-levered belt joins §5’s kinked cords in the pile of drives that failed weeks after they were “fixed”. Tension to the card, re-tension after the first day’s bedding-in, and the exponential does the rest.

§7Quick reference

The working core of the page on one card rack.