Mechanical Power Transmission, V-Belt Sheave Drives, Pitch Length & Wrap Angle
V-Belt Sheave Pitch Length & Small-Pulley Wrap Formula
Calculate Open-Drive V-Belt Pitch Length Lp (mm), Small-Sheave Arc of Contact Wrap Angle (deg), Speed Reduction Ratio (D/d), and Linear Belt Velocity (m/s).
Shorten a V-belt motor mount to save floor space and small-sheave wrap drops below 150°—slashing kW belt capacity by 12%–20%!
When coupling an induction motor Small Sheave `d` (mm) to a fan, crusher, or compressor Large Sheave `D` (mm) across Shaft Center Distance `C` (mm), two geometric effects happen simultaneously that naive `2*C + PI()*(D+d)/2` estimates miss: (1) Diagonal belt spans add quadratic Pitman/Rankine slack term **`((D - d)^2) / (4 * C)`** to **Exact Open-Belt Pitch Length `L_p = 2 * C + (PI() / 2) * (D + d) + ((D - d)^2) / (4 * C)`** (mm), AND (2) pulley diameter asymmetry steals wrap angle from the motor pulley, reducing **Small-Sheave Arc of Contact `theta_1 = 180° - 2 * ASIN((D - d) / (2 * C)) * (180 / PI())`**! Under RMA / ISO 4184 V-belt rating tables (`180°` wrap = `C_theta = 1.00`), letting `theta_1` fall to `140°` derates power transmission (`C_theta ≈ 0.89`) and invites belt squeal and glazed sidewalls—while **Linear Belt Speed `v = (PI() * (d / 1,000) * RPM) / 60`** must stay below `30–33` m/s (`~6,000–6,500` fpm) to avoid centrifugal tension throw-off on cast-iron sheaves!
Verify on a `1,750` RPM Industrial Blower V-Belt Drive—Driven_Large_Sheave `D` = 315.0 mm (B2), Motor_Small_Sheave `d` = 125.0 mm (B3 -> Speed Ratio = `2.52` in E4), Slide-Base_Center_Distance `C` = 520.0 mm (B4), and Motor_Speed = 1,750 RPM (B5): Belt Pitch Length equals `1,748.5` mm (E2), Small-Sheave Wrap Angle holds `158.95°` (`C_theta ≈ 0.95` in E3), and Linear Belt Speed runs cleanly at `11.45` m/s (E5)! In a shared plant-maintenance workbook, a millwright with `suggester` access can propose a tighter slide-base center distance while an engineering `owner` or `editor` checks `theta_1` before accepting the cell suggestion.
| ISO 4184 V-Belt Drive KPI (D=315 mm, d=125 mm, C=520 mm, 1,750 RPM) | Formula | Verified Output |
|---|---|---|
| V-Belt Pitch Length Lp (mm) | 2*C + (PI()/2)*(D+d) + ((D-d)^2)/(4*C) | 1,748.5 mm (1,283.4 mm @ short C=280 mm) |
| Small-Sheave Wrap Angle (deg) | 180 - 2 * ASIN((D-d)/(2*C)) * (180/PI()) | 158.95° (140.33° slip risk @ C=280 mm!) |
| Speed Reduction Ratio (D / d) | Large_Sheave_D_mm / Small_Sheave_d_mm | 2.52 (:1) |
| Linear Belt Speed v (m/s) | (PI() * (d_mm / 1000) * Motor_RPM) / 60 | 11.45 m/s (well below 30 m/s iron limit) |
Open-Drive V-Belt Pitch Length, Small-Pulley Arc of Contact & Rim Speed Formula
Validate `Small_Sheave d > 20` mm, `Large_Sheave D >= d`, non-touching sheaves `Center_Distance C > (D + d) / 2`, and `Motor_RPM > 0`! Try building the arc-of-contact `ASIN` conversion from radians to degrees first; if your wrap angle looks near `179°`, ask the in-app AI assistant for a hint on scaling `ASIN(...)` by `180 / PI()` rather than pasting a blind answer.
=LET(dl_m, B2, ds_m, B3, cd_m, B4, nr_m, B5, vs_m, (PI() * (ds_m / 1000) * nr_m) / 60, IF(OR(ds_m <= 20, dl_m < ds_m, cd_m <= (dl_m + ds_m) / 2, nr_m <= 0), "Invalid Sheave Drive", ROUND(vs_m, 2)))Cramped Skid 280 mm Center-Distance Wrap-Loss Check (Set Center Distance B4 = 280 mm)
Slide the motor base inward to `C` B4 = 280 mm (`< D = 315` mm rule-of-thumb minimum!): Pitch Length E2 drops to `1,283.4` mm and Small-Sheave Wrap Angle E3 plunges from `158.95°` down to `140.33°`—requiring an extra belt groove or grooved backside idler to prevent startup slip!
=IF(ISTEXT(E5), E5, ROUND(180 - 2 * (ASIN((B2 - B3) / (2 * B4)) * (180 / PI())), 2))Try the example
Use the original values or change them to test your understanding.