Belt, Clutch and Brake Torque: Method

Friction Drives and Brakes: Belts, Clutches, Shoes and Bands

Belts, clutches and brakes all work by dry friction between two surfaces pressed together, and the same few relations of classical mechanics describe them: Coulomb's law f = F_friction/N on every element of contact, Euler's exponential law for a flexible member wrapped on a cylinder, and the integrals of pressure times friction over a disc, a cone or a shoe. This page gives the theory behind the three analyses of the tool, the sign conventions of its figures, and what the results mean. Nothing in it is a code rule; the pressure limits and coefficients of friction are the analyst's inputs.

Belt drive

Geometry. Two pulleys of pitch diameters D₁ (driver) and D₂ at the centre distance C. For an open belt the straight runs are tangent to both circles on the same side and make the angle ψ with the centre line, sin ψ = (D₂ − D₁)/(2C); the belt wraps the smaller pulley through θ₁ = π − 2ψ and the larger through θ₂ = π + 2ψ, and its length is exactly L = √(4C² − (D₂ − D₁)²) + (θ₁D₁ + θ₂D₂)/2. For a crossed belt the runs are the internal tangents, sin ψ = (D₁ + D₂)/(2C), both wraps are π + 2ψ and L = √(4C² − (D₁ + D₂)²) + θ(D₁ + D₂)/2. The familiar series form L ≈ 2C + π(D₁ + D₂)/2 + (D₂ ∓ D₁)²/(4C) is the expansion of the exact form in (D/C)²; the tool prints both. The belt speed is v = ω₁D₁/2 and, with no slip, the driven pulley turns at n₂ = n₁D₁/D₂.

Euler's relation. Consider an element of belt subtending dθ on a pulley, with tension F on one end and F + dF on the other. The normal force from the pulley is F dθ and, when the belt is about to slip, the friction on the element is f F dθ, which is also the tension difference dF. So dF/F = f dθ and, integrating over the wrap, F₁/F₂ = e^fθ: the tight-side tension can exceed the slack-side tension by at most this ratio before the belt slides. Euler published this law in 1762 for a rope on a fixed cylinder (the "Remarques sur l'effet du frottement dans l'équilibre"); it is the capstan equation of seamanship and is written in the same form, T₁/T₂ = exp(μφ), in the NASA-contract belt study of 1966 cited by the tool.

Centrifugal tension. A belt of mass m′ per unit length moving at v round a pulley needs a centripetal force m′v² dθ on each element, which the tension supplies: the part F_c = m′v² of the tension does not press the belt on the pulley and does not take part in the friction. Euler's law then applies to the excess over F_c: (F₁ − F_c)/(F₂ − F_c) = e^fθ. The tool takes the belt weight per length w as entered and uses m′ = w/g. At high speed F_c eats the available tension, which is why the power a belt can carry at a given installed tension peaks at v = √(F_i/(3m′)) and falls to zero when F_c reaches F_i.

V belts. In a V groove of included angle 2β the belt sits on its flanks; a radial force N pressing it into the groove produces a normal force N/(2 sin β) on each flank, so the friction available per unit radial force is f/sin β. Every relation for the flat belt holds with f replaced by this effective value: a 38° groove with f = 0.30 behaves like a flat belt with f = 0.92. The tool applies the correction to the governing (smaller) wrap; it does not model a V belt running on a flat pulley.

Tensions for a load. The transmitted power fixes the tension difference F₁ − F₂ = P/v = 2T₁/D₁. Two problems follow. If the belt is to be installed just tight enough to carry the load, the friction is fully used on the pulley with the smaller wrap: F₁ − F_c = (F₁ − F₂)e^fθ/(e^fθ − 1), F₂ = F₁ − (F₁ − F₂), and the installed tension is the mean F_i = (F₁ + F₂)/2 (the belt is tensioned at rest and the two sides change by equal and opposite amounts under load, so their mean stays at the installed value; this is also the form of the 1966 JPL study). If instead the installed tension is known, F₁ = F_i + (F₁ − F₂)/2, F₂ = F_i − (F₁ − F₂)/2, and the belt holds as long as the actual ratio (F₁ − F_c)/(F₂ − F_c) stays below e^fθ; the largest tension difference it can carry is 2(F_i − F_c) tanh(fθ/2), which the tool converts to the largest power at the entered speed. The chart draws the tensions at impending slip against power so that the margin of an installed tension can be read.

Stresses. With the belt section A entered the tool prints the tight-side stress F₁/A and the centrifugal part F_c/A; with the thickness t and modulus E it adds the bending stress E t/D₁ on the small pulley, which is the outer-fibre strain t/D₁ of a belt bent to the pulley radius times the modulus (the belt's neutral axis at mid-thickness). The largest stress, F₁/A + E t/D₁, occurs where the tight side bends round the small pulley and governs belt fatigue; manufacturers' ratings fold it into their horsepower tables, which this tool does not reproduce.

Disc and cone clutches

Two assumptions. A friction annulus from r_i to r_o is pressed by the axial force F; the pressure p(r) depends on how the lining has worn. A new, rigid clutch presses evenly: uniform pressure, p = p_a. A clutch that has run in wears each ring in proportion to the work of friction there, f p v, and since v = ωr the wear is uniform only if p r = constant: uniform wear, p = p_ar_i/r, with the largest pressure at the inner radius. Integrating the pressure over the annulus, dA = 2πr dr, gives the force and the torque of n friction faces:

uniform wear: F = 2π p_a r_i (r_o − r_i), T = n f F (r_o + r_i)/2;
uniform pressure: F = π p_a (r_o² − r_i²), T = n f F · 2(r_o³ − r_i³)/(3(r_o² − r_i²)).

The quantity multiplying n f F is the effective friction radius: the mean radius under uniform wear, slightly more under uniform pressure. For the same pressure limit the uniform-wear capacity is lower (0.79 of the other at r_i/r_o = 2/3), and it is the usual design assumption because a clutch spends its life worn in. The two values bracket the truth; the tool prints both and their ratio. Under uniform wear at a fixed outer radius and pressure limit the capacity f n π p_a r_i(r_o² − r_i²) is largest at r_i = r_o/√3 = 0.577 r_o, which the chart marks; a narrower annulus wastes outer area, a wider one wastes pressure at the inside.

Cone clutch. If the friction surface is a cone of half angle α (the angle between the surface and the axis), an element of the surface is dA = 2πr dr/sin α. The axial force is the normal pressure times sin α integrated over the surface, so F is exactly the disc value for the same radii, while the normal force on the surface is F/sin α and the torque is the disc torque divided by sin α. With α = 12° the same axial force gives 4.8 times the torque of a flat annulus, which is the point of the cone; the price is the large normal force and the wedging that may make the clutch hard to release. α = 90° is the disc.

Required torque. When the torque is given, the force and pressure follow by proportion from the pressure-limit solution, and the tool compares the pressure needed with the limit under both assumptions. A multiple-disc clutch counts one friction face per rubbing pair; the axial force is the same on every face.

Drum brakes with shoes

Frame of the figure. The drum of radius r turns about O. A shoe is hinged at the pin A on the line through O, at the distance a from O: inside the drum (a < r) for an internal expanding shoe, outside (a > r) for an external contracting shoe. The lining spans the angles θ₁ to θ₂ measured from the line O–A toward the shoe, the heel near the pin and the toe at θ₂. The actuating force F_a presses the shoe onto the drum, acting across the pin line at the toe end with the moment arm c about the pin. The tool draws the internal shoe with the pin below O and the external shoe with the pin above O, the shoe on the right in both cases, and the rotation sense is read on that figure.

Pressure along a long shoe. As the shoe rotates slightly about its pin, a point of the lining at θ moves normal to the drum by an amount proportional to its distance from the pin line, a sin θ; the lining wears in proportion, and the pressure that results is p = p_a sin θ/sin θ_a, largest at θ_a = 90° (or at θ₂ if the shoe ends before 90°). This is the long-shoe theory of Shigley's Chapter 16 and of every machine-design text; the tool derives its moments from the force vectors rather than quoting them.

Moments about the pin. On an element b r dθ of lining act the normal force p b r dθ along the radius and the friction f p b r dθ along the drum surface in the direction of the drum's motion. Taking moments about the pin and integrating from θ₁ to θ₂:

M_N = (p_a b r a/sin θ_a) ∫ sin²θ dθ = (p_a b r a/sin θ_a) [(θ₂ − θ₁)/2 − (sin 2θ₂ − sin 2θ₁)/4],
M_f = (f p_a b r/sin θ_a) ∫ sin θ (r − a cos θ) dθ = (f p_a b r/sin θ_a) [r(cos θ₁ − cos θ₂) − a(sin²θ₂ − sin²θ₁)/2],
T = ∫ f p b r² dθ = f p_a b r² (cos θ₁ − cos θ₂)/sin θ_a.

The normal forces always turn the shoe off the drum, against the actuating force. The friction forces turn it one way or the other depending on the rotation: if the drum surface drags the lining in the sense that presses the shoe harder onto the drum, the friction moment helps the actuating force and the shoe is self-energizing, F_ac = M_N − M_f; with the opposite rotation it is de-energizing, F_ac = M_N + M_f. For the internal shoe of the figure (pin below O, shoe on the right) clockwise rotation is self-energizing: the surface moves from the toe toward the heel and drags the shoe into the drum, the wedging that the Army's TM 9-8000 describes for automotive brakes, where the primary shoe self-energizes and, through the link to the secondary shoe, produces the servo action. For the external shoe of the figure (pin above O) the sense is reversed because the normal force is reversed. The tool states which it is for the rotation chosen; a two-shoe brake with one shoe of each kind gets the sum of the two torques for one actuating force on each.

Self-locking. A self-energizing shoe locks when M_f reaches M_N: the brake then grabs with no actuating force and cannot be released by the lever. The tool prints the coefficient of friction f_lock = f M_N/M_f at which that happens for the geometry entered; designs keep the working f well below it because f rises with wear, moisture and temperature. The same condition fixes the pin position: moving the pin toward the drum surface (a → r) shrinks the friction arm r − a cos θ and raises f_lock.

Capacity and entered force. At the pressure limit p_a every force is fixed and the tool reports the actuating force needed and the torque obtained. If the actuating force is entered instead, all the forces scale in proportion (the pressure distribution keeps its shape), so the tool reports the pressure reached and the torque; a pressure above the limit is flagged. The hinge-pin reaction follows from force equilibrium of the shoe, R = −(F_a + N + F_f), with the components given in the figure's axes.

Short shoe. When the lining is narrow (an arc of a few tens of degrees) the pressure is taken uniform and the normal force N = p_a b r(θ₂ − θ₁) and the friction f N are concentrated at the centre θ_c of the arc; the torque is f N r and the moments about the pin are N a sin θ_c and f N (r − a cos θ_c). The long-shoe integrals reduce to these as the arc shrinks, which the tool's checks verify; for arcs beyond about 45° the long-shoe analysis should be used.

Band brakes

A flexible band of width b wrapped through φ on the drum obeys Euler's relation with the drum turning toward the tight end: P₁ = P₂e^fφ. The torque is (P₁ − P₂)r, and the pressure under the band is p = P/(b r), largest at the tight end, where it is compared with the lining limit. The band ends are pulled by a lever pivoted at the drum side: the slack end at the arm a₂ and, in a differential brake, the tight end at the arm a₁ on the other side of the pivot, so that the moment balance is F L = P₂a₂ − P₁a₁. With a₁ = 0 (the simple band, tight end anchored at the pivot) the operating force is P₂a₂/L, small because the lever pulls the slack end only; reversing the drum would make the lever pull the tight end and need e^fφ times the force, which is why the simple band brakes one way. With a₁ > 0 the tight end helps the lever: the brake is self-energizing and locks when a₂ ≤ a₁e^fφ, the condition under which it becomes a backstop rather than a brake. The Navy's Basic Machines describes the band of anchor and cargo winches, tightened by a screw instead of a lever, as the common shipboard form.

Energy and heat of a stop

A clutch bringing a load up to speed or a brake stopping it converts the kinetic energy of the rotating parts, E = I(ω₀² − ω_f²)/2, into heat at the friction surface. With the torque T taken constant through the slip the time is t = I(ω₀ − ω_f)/T, the energy equals T times the angle turned, the mean power is E/t and the power at the first instant T ω₀ (twice the mean for a stop to rest). If the drum or plate of mass m and specific heat c absorbs the heat of one stop before any of it is conducted or convected away, its temperature rises by E/(m c); the figure is an upper bound for a single stop and a lower bound for repeated stops, whose heat accumulates until the cooling of the housing balances the mean power. Lining ratings are usually quoted as a pressure–velocity product or an energy per unit area per stop, which the analyst compares with the values printed.

Reading the results and limits

The tool prints each quantity with its equation and the numbers substituted, the figure of the configuration as entered, and one chart per analysis: belt tensions against power, clutch capacity against the radius ratio, the lining pressure along a long shoe, the band ratio against the wrap angle. The checks behind it compare every closed form with a numerical integration of the defining integrals and of the force vectors (30 random shoe geometries in both frames and both senses of rotation), and reproduce the Prony-brake example of the Navy manual. Not covered: belt creep and the slip-dependent efficiency of real drives, manufacturers' belt ratings and service factors, temperature-dependent friction, the dynamics of engagement (torque varying with slip speed), two-shoe brakes with the shoes linked (servo brakes) beyond adding the two torques, disc brakes with pad geometry other than an annular sector (use the annulus with the pad's angular fraction), and the thermal transient of repeated stops.