How a Belt Drive Transmits Torque

Why this matters

Most techs treat a belt as a rope that turns a fan. That model gets you through a straight replacement and fails you everywhere else: it cannot tell you why a drive that ran for six years started slipping the week nothing was touched, why a wider belt did not fix it, or why the fix that worked was a bigger small sheave rather than more tension. A belt does not carry torque because it is strong. It carries torque because friction lets the two sides of the loop hold different tensions, and that difference is bounded by geometry you can measure with a tape and a protractor. Once you can compute the bound, the diagnosis stops being a guess.

Isolate before you get near the drive

Lock out and tag the motor at its disconnect before your hands go inside the guard. The rotational energy stored in a fan wheel or a flywheel is mechanical stored energy under 29 CFR 1910.147, so power off is not the same as safe: verify zero rotation by eye, and where the wheel sits in a duct with any air moving through it, close and secure the damper or physically block the wheel, because a windmilling wheel will drive the sheave backwards through the belt. If you must open the electrical enclosure to reach the disconnect, that is electrical work under 29 CFR 1910.333(b)(2), and prove your meter live-dead-live per NFPA 70E-2021, 120.5 before you trust a zero reading. The guard goes back on before restart: guarding of belts, pulleys and sheaves is required by 29 CFR 1910.219 in general industry and 29 CFR 1926.300(b) on a construction site.

A belt can only pull

A belt has no meaningful compressive or bending strength in the direction of travel. The only force it can apply to a sheave is tension along its own length. So every question about torque capacity reduces to one question: how much more tension can the incoming side hold than the outgoing side, before the belt stops gripping and starts sliding.

Name the two sides properly, because the labels move with rotation and half the confusion in the field comes from mixing them up. The tight side is the span running into the driver sheave's departure point and pulling on the driven sheave. The slack side is the return span. Reverse the motor and the two swap ends. Torque delivered at either sheave is:

torque = (tight-side tension - slack-side tension) x effective radius

That difference is called the net pull, and it is the same value at both sheaves. What differs between the two ends is the radius it acts on, which is where the ratio comes from.

Two consequences follow immediately. First, the total tension in the loop does not appear in that equation at all: you can double both tensions and transmit exactly the same torque, while doubling the load on every bearing in the drive. Second, a belt with zero slack-side tension transmits zero torque, because there is nothing left to subtract from.

Why the two sides can hold different tensions

Around the arc of contact, each small element of belt is held by friction against the sheave. Walk from the slack side to the tight side and tension builds a little at each element, because each element's friction adds to what the element behind it was already holding. The build-up is multiplicative, not additive, which is why the limit is an exponential.

For a flat belt on a round pulley, treating the belt as perfectly flexible and inextensible, with a uniform coefficient of friction, at impending slip and ignoring centrifugal effects, the limiting ratio is the capstan relation:

tight tension / slack tension = e^(mu x theta)

with theta the wrap angle in radians and mu the coefficient of friction. Every one of those conditions matters. A stiff cogged belt is not perfectly flexible. A glazed or oil-wetted sidewall does not have the mu you assumed. And the relation gives the limit at the point of slipping, not the ratio your drive actually runs at.

Wrap angle on the small sheave, which is always the limiting one, is set by the diameters and the center distance:

theta = 180 degrees - 2 x arcsin((D - d) / (2C))

with D the large pitch diameter, d the small pitch diameter and C the center distance, all in the same unit. Pull the motor closer and you lose wrap on the small sheave. That is the mechanism behind the classic field result where sliding the motor in to get slack ends up making a drive slip worse than it did.

What the V-groove actually adds

A V-belt does not grip better because it is rubber. It grips better because the groove wedges it. As the belt is pulled radially into a groove of included angle beta, the two flanks each push back with a normal force larger than the radial load by a factor of 1/sin(beta/2). Friction acts on that larger normal force, so the belt behaves as if it had an effective coefficient:

mu_effective = mu / sin(beta / 2)

Industrial V-sheave grooves commonly run in the 34 to 38 degree included range, with smaller pitch diameters using the smaller angle so the belt does not jam as it bends into the groove, and the exact angle for a given section and diameter is published by the sheave maker. Take a 36 degree groove: sin(18 degrees) is 0.309, so the wedge multiplies the effective coefficient by about 3.2. That is the whole reason a V-belt in a 4 inch sheave does work that a flat belt of the same width could not.

It also tells you what destroys capacity. The wedge depends on the belt contacting both flanks, not the groove bottom. A belt riding on the bottom of the groove has no wedge at all, and its capacity collapses to something near a flat belt's. That failure has its own article; see the reference on what a worn sheave groove does to a new belt.

Worked example: a fan drive from horsepower to belt pull

Take a supply fan. Motor nameplate 1750 rpm, 3.0 hp. Driver sheave pitch diameter 4.0 in, fan sheave pitch diameter 12.0 in, center distance 18.0 in.

Speed. Fan speed is 1750 x 4.0 / 12.0 = 583 rpm.

Torque at the motor shaft. Using the standard conversion for inch-pound units, torque in lb-in equals 63,025 x hp / rpm, so 63,025 x 3.0 / 1750 = 108 lb-in.

Net pull. The effective radius at the driver is half its pitch diameter, 2.0 in. Net pull = 108 / 2.0 = 54 lb. That single number is what the drive has to produce through friction, and it is the same 54 lb acting at the 6.0 in radius of the fan sheave, giving 324 lb-in there. Speed fell by three, torque rose by three, and 54 lb is the currency both ends trade in.

Splitting it into two tensions. A design tension ratio of 5 is a common working figure for V-belt drives, chosen well inside the slip limit. With tight over slack equal to 5 and their difference equal to 54 lb, the slack side is 54 / (5 - 1) = 13.5 lb and the tight side is 67.5 lb.

Shaft side load. Wrap angle on the small sheave is 180 - 2 x arcsin(8.0 / 36.0) = 180 - 2 x 12.8 = 154 degrees. That leaves the two spans 26 degrees from parallel, so the shaft load is the vector sum of 67.5 and 13.5 lb at that included angle, about 80 lb, against an arithmetic sum of 81 lb. On this geometry the two are within about 1 percent and you can use the sum; on a drive with a much smaller wrap they diverge and you must resolve the vectors. What that 80 lb does to the bearings either side is covered in its own reference.

How much margin is there? Convert 154 degrees to radians: 2.69. Take an illustrative dry rubber-on-cast-iron coefficient near 0.3, the figure commonly used for clean flat contact at moderate speed, and wedge it through a 36 degree groove: 0.3 / 0.309 = 0.97. The capstan limit is then e^(0.97 x 2.69) = e^2.61, about 13.6. The drive is designed at a ratio of 5 against a slip limit near 13.6, so it is running at roughly a third of its friction capacity. That margin is why the drive tolerates a dirty morning and a cold start. It is also why a drive that slips has usually had something specific taken away from it: mu cut by oil or glaze, theta cut by a moved motor, or the required net pull raised by a load change.

The term that quietly eats the margin. Belt speed here is pi x 4.0 x 1750 / 12 = 1,832 ft/min, or 30.5 ft/s. Centrifugal tension is (weight per foot / g) x velocity squared, and it does not help transmit anything: it pushes the belt out of the groove and subtracts from the normal force available for friction. At about 0.08 lb per foot for a small industrial section, that is (0.08 / 32.2) x 30.5^2 = 2.3 lb, against a 13.5 lb slack side. Ignorable here. Run the same belt at 5,000 ft/min and the term scales with the square of speed, (5000 / 1832)^2 = 7.4 times, or roughly 17 lb, which is larger than this drive's entire slack side. That is why high-speed drives need more installed tension for the same torque, and why a sheave change that raises belt speed can make a previously stable drive slip without anything else changing.

Verifying this on a machine in front of you

Measure, do not assume, and take all three of these with the drive locked out and the wheel restrained:

• Center distance and both pitch diameters, so you can compute the wrap angle rather than eyeball it. Anything under about 120 degrees on the small sheave is a drive with little friction margin left, and an idler that increases wrap is a legitimate fix where moving the motor is not.
• Which span is tight under load. Bar the drive over by hand from the sheave rim, never by gripping a span, and confirm the tight side is the one your torque direction predicts. A drive installed with the adjustable idler on the tight side is a drive fighting itself.
• Groove and belt fit. The belt top should sit flush with or slightly above the sheave rim. Riding low means the wedge is partly gone, and no amount of tension restores it.

Then close the loop with numbers: compute the predicted driven speed from pitch diameters, read the actual with an optical tachometer from outside the guard with the guard in place, and treat any gap beyond about 1 percent as a real finding rather than instrument error. What that gap means, and what it costs, is the subject of the companion article on slip.

References

• 29 CFR 1910.147, control of hazardous energy, for mechanical isolation and stored rotational energy
• 29 CFR 1910.333(b)(2) for electrical work at the disconnect, and NFPA 70E-2021, 120.5 for the live-dead-live proving sequence
• 29 CFR 1910.219 (general industry) and 29 CFR 1926.300(b) (construction) for guarding of belts, pulleys and sheaves
• Belt and sheave manufacturer engineering data for groove angles by section and pitch diameter, published pitch diameters, and design tension ratios
• See related: What Belt Tension Actually Controls; Why a Belt Slips and What It Costs; What a Worn Sheave Groove Does to a New Belt