Belt Drive Calculator

Open Belt Length (second-order approximation)

L = 2·C + (π/2)·(D₁ + D₂) + (D₁ − D₂)² / (4·C)

Variables: L = belt pitch length (m); C = centre distance between shaft axes (m); D₁ = driver pulley pitch diameter (m); D₂ = driven pulley pitch diameter (m). This is the standard second-order approximation to the exact open-belt length, not the exact closed form. Verified error is 2.2×10⁻⁴ % at the tool's defaults and 0.02 % at the design-rule minimum C = (D₁ + D₂)/2, so it is accurate to far better than 0.1 % over the whole recommended range (D₁ + D₂)/2 ≤ C ≤ 3·(D₁ + D₂) that the tool prints as its centre-distance design rule. Source: Shigley's Mechanical Engineering Design, 10th ed., Ch. 17-3 'V Belts' (approximate pitch-length / centre-distance relation); the same approximation for open flat-belt drives appears in Khurmi & Gupta, Theory of Machines, ch. 'Belt, Rope and Chain Drives'. Note that Shigley's flat-belt section (17-2) instead gives the exact form L = √(4·C² − (D₁ − D₂)²) + (D₁·θ_large + D₂·θ_small)/2.

Angles of Wrap (open drive)

θ_w = π − 2·asin( (D₁ − D₂) / (2·C) ) | ← what the tool evaluates and labels 'small pulley' | Correct general form: | θ_small = π − 2·asin( |D₁ − D₂| / (2·C) ) | and | θ_large = π + 2·asin( |D₁ − D₂| / (2·C) ) | with | θ_small + θ_large = 2·π

Variables: θ_w = the wrap angle the tool computes and uses (rad; also displayed in degrees); θ_small = angle of wrap on the SMALLER pulley (rad); θ_large = angle of wrap on the LARGER pulley (rad); D₁ = driver pulley diameter (m); D₂ = driven pulley diameter (m); C = centre distance (m). The asin argument is dimensionless and is clamped to −1…+1 in the code. IMPORTANT: the tool omits the absolute value, so θ_w equals θ_small only when D₁ ≥ D₂. For a speed reducer (driver smaller than driven, D₁ < D₂ — the common case) θ_w is the LARGER pulley's wrap, mislabelled. Worked check at D₁ = 0.15 m, D₂ = 0.30 m, C = 0.8 m: the tool displays 190.76° as the 'small pulley' wrap when the true small-pulley wrap is 169.24°. The design note recommends θ_small ≥ 120°. Source: Shigley's Mechanical Engineering Design, 10th ed., Ch. 17-2 'Flat- and Round-Belt Drives' (open-belt contact angles θ_d on the small sheave and θ_D on the large sheave)

Capstan (Eytelwein) Tension Ratio and Belt Tensions

F₁ / F₂ = e^(μ·θ_w) | F_e = P / v = F₁ − F₂ | F₁ = F_e · e^(μ·θ_w) / (e^(μ·θ_w) − 1) | F₂ = F₁ / e^(μ·θ_w)

Variables: F₁ = tight-side tension (N); F₂ = slack-side tension (N); F_e = effective (net driving) tension (N); μ = belt-to-pulley coefficient of friction (dimensionless; 0.3–0.5 for flat rubber); θ_w = the wrap angle the tool computes (rad — see the wrap-angle entry; slip is governed by the SMALLER pulley's wrap, which θ_w only equals when D₁ ≥ D₂); P = transmitted power (W); v = belt linear speed (m/s). The product μ·θ_w is dimensionless because θ_w is in radians. Two limits of this model: (a) it is the FLAT-belt exponent, so a V-belt requires the effective friction μ/sin(β/2) with β = groove angle (deg) — entering the 0.5–0.7 V-belt μ that the input helper text suggests under-predicts the tension ratio and over-predicts F₁; (b) centrifugal tension is neglected, the full relation being (F₁ − F_c)/(F₂ − F_c) = e^(μ·θ_small) with F_c = m′·v² (F_c in N, m′ = belt mass per unit length in kg/m), which is non-conservative above roughly 20–25 m/s — the tool's own defaults already sit at v = 23.56 m/s. Source: Shigley's Mechanical Engineering Design, 10th ed., Ch. 17-2 'Flat- and Round-Belt Drives' (belt-tension relations including centrifugal tension); the underlying capstan relation is the Euler–Eytelwein equation

Belt Speed and Speed Ratio

v = π · D₁ · n₁ / 60 | Correct kinematics (equal surface speeds): | n₁ · D₁ = n₂ · D₂ | → | n₂ = n₁ · D₁ / D₂ | The tool instead evaluates | i = D₁ / D₂ | and | n₂ = n₁ / i = n₁ · D₂ / D₁

Variables: v = belt linear speed (m/s); D₁ = driver pulley pitch diameter (m); D₂ = driven pulley pitch diameter (m); n₁ = driver shaft speed (rev/min); n₂ = driven shaft speed (rev/min); i = the ratio the tool displays as 'Speed Ratio i' (dimensionless) — it is the DIAMETER ratio D₁/D₂, not a speed ratio. The belt-speed result v is correct and is what drives F_e, F₁ and F₂, so only the displayed n₂ is affected. The displayed n₂ is wrong by a factor (D₁/D₂)²: at the shipped defaults (D₁ = 0.3 m, D₂ = 0.15 m, n₁ = 1500 rev/min) the tool reports 750 rev/min where the correct value is 3000 rev/min. Source: Shigley's Mechanical Engineering Design, 10th ed., Ch. 17-2 'Flat- and Round-Belt Drives' (belt-drive kinematics and belt velocity, V = π·d·n/60 in SI units)