MEC-43

Belt ratio

Euler capstan formula.

Reading speed

MachinesBelt ratio

Governing equation

F_1/F_2=e^{\mu\theta}

$\theta \in \left[1.5, 4\right]$

where

F_2: Slack (N)
\mu: Friction (—)
\theta: Wrap (rad)
F_1: Tight (N)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

F_1/F_2=e^{\mu\theta}

$\theta \in \left[1.5, 4\right]$

$\theta = 1.5$ $F_1 = 313.7$

Along the curve

②

Domain of the variable — blue

\theta_{\min} = 1.5,\quad \theta_{\max} = 4

$\theta \in \left[1.5, 4\right]$

$\theta = 1.5$ $\theta = 1.5$

Along the curve

③

Response curve — green

F_1/F_2=e^{\mu\theta}

$\theta \in \left[1.5, 4\right]$

$\theta = 1.5$ $F_1 = 313.7$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}F_1}{\mathrm{d}\theta}

$\theta \in \left[1.5, 4\right]$

$\theta = 1.516$ $F_1^{\prime} = 94.54$

Along the curve

Lecture brief

Historical brief

Machine design grew from Coulomb torsion and Hertz contact (1881) through Soderberg fatigue and the heat-engine cycle. The lab writes shaft, bearing, contact and thermodynamic limits in SI. This sheet (MEC-43 — Belt ratio) is the form associated with Belt ratio. Working symbols: $F_2$ , $\mu$ , $\theta$ $\rightarrow$ $F_1$ . Euler capstan formula. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

F_1

from

F_2

\mu

\theta

in Mechanical via

F_1/F_2=e^{\mu\theta}

Euler capstan formula. Use it when a real mechanical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

F_2 = 200.000\,\mathrm{N}

\mu = 0.300\,\mathrm{—}

\theta = 2.800\,\mathrm{rad}

, the governing relation

F_1/F_2=e^{\mu\theta}

yields

F_1 = 463.3\,\mathrm{N}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Slack F_2200 NFriction \mu0.3 —Wrap \theta2.8 rad

Outputs

Tight F_1463.3 N

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Euler capstan formula. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube