Capstan and Flat-Belt Friction Formula (Euler-Eytelwein Equation)

Catch the degrees-vs-radians bug inside EXP(mu * theta_rad)!

Euler and Eytelwein proved that normal contact pressure along a flexible rope, winch drum, or flat drive belt grows exponentially with wrap angle theta_rad (in RADIANS, completely independent of drum radius!): at impending slip, tight-side tension T_1 to slack-side tension T_2 satisfies T_1 / T_2 = EXP(mu * theta_rad)! Passing wrap angle in degrees (such as 180 instead of PI() = 3.1416) inside EXP(0.30 * 180) explodes the tension ratio by 23 orders of magnitude!

Verify two engineering regimes using RADIANS(theta_deg): (1) a 180-degree pulley drive (theta_deg = 180 in B3 -> PI() rad, friction mu = 0.35 in B2, slack tension T_2 = 400 N in B4, belt speed v = 12 m/s in B5) has tension ratio EXP(0.35 * PI()) = 3.0028, max tight tension T_1 = 1,201.1347 N, and transmitted power (1201.1347 - 400) * 12 / 1000 = 9.6136 kW; (2) wrapping 3 full turns (1,080 degrees = 6*PI() rad) around a marine capstan multiplies a 400 N tailing pull by EXP(0.35 * 6 * PI()) = 733.1458x!

Euler-Eytelwein Tension Amplification Ratio EXP(mu * RADIANS(theta_deg)) Formula

Convert degrees to radians explicitly via RADIANS(B3) inside LET so users entering 180, 210, or 1080 degrees always receive exact dimensionless amplification.

=LET(mu_f, B2, deg_w, B3, IF(OR(mu_f < 0, deg_w <= 0), "Invalid Wrap Input", ROUND(EXP(mu_f * RADIANS(deg_w)), 4)))

Maximum Slip-Free Belt Power P_kW = T_2 * (EXP(mu * theta) - 1) * v / 1000 Formula

Notice that frictionless contact (set mu = 0 in B2) collapses EXP(0) - 1 to 0, reducing transmitted power cleanly to 0.0000 kW.

=LET(mu_f, B2, deg_w, B3, t_slk, B4, v_blt, B5, IF(OR(mu_f < 0, deg_w <= 0, t_slk <= 0, v_blt < 0), "Invalid Wrap Input", LET(t_rat, EXP(mu_f * RADIANS(deg_w)), ROUND((t_slk * (t_rat - 1) * v_blt) / 1000, 4))))

Further reading

• Arte Sheets Lesson — Lewis Gear-Tooth Bending Stress and Barth Velocity Factor
• Microsoft Support — RADIANS function