MEC-30

Gear ratio and torque

ω2/ω1 = N1/N2, T2/T1 = η N2/N1. Teeth count sets the reduction.

Reading speed

GearsWillis

Governing equation

\dfrac{\omega_2}{\omega_1}=\dfrac{N_1}{N_2},\quad \dfrac{T_2}{T_1}=\eta\dfrac{N_2}{N_1}

$N_2 \in \left[12, 200\right]$

where

N_1: Pinion teeth (—)
N_2: Wheel teeth (—)
\omega_1: Pinion speed (rad/s)
T_1: Pinion torque (N·m)
\eta: Mesh efficiency (—)
\omega_2: Wheel speed (rad/s)
T_2: Wheel torque (N·m)

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

\dfrac{\omega_2}{\omega_1}=\dfrac{N_1}{N_2},\quad \dfrac{T_2}{T_1}=\eta\dfrac{N_2}{N_1}

$N_2 \in \left[12, 200\right]$

$N_2 = 12$ $\omega_2 = 180$

Along the curve

②

Domain of the variable — blue

N_2_{\min} = 12,\quad N_2_{\max} = 200

$N_2 \in \left[12, 200\right]$

$N_2 = 12$ $N_2 = 12$

Along the curve

③

Response curve — green

\dfrac{\omega_2}{\omega_1}=\dfrac{N_1}{N_2},\quad \dfrac{T_2}{T_1}=\eta\dfrac{N_2}{N_1}

$N_2 \in \left[12, 200\right]$

$N_2 = 12$ $\omega_2 = 180$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\omega_2}{\mathrm{d}N_2}

$N_2 \in \left[12, 200\right]$

$N_2 = 13.18$ $\omega_2^{\prime} = -12.53$

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-30 — Gear ratio and torque) is the form associated with Willis. Working symbols: $N_1$ , $N_2$ , $\omega_1$ , $T_1$ , $\eta$ $\rightarrow$ $\omega_2$ , $T_2$ . A meshing constraint: arc speeds match at the pitch point. Efficiency η < 1 accounts for tooth friction.

Purpose

Purpose: compute

\omega_2

T_2

from

N_1

N_2

\omega_1

T_1

\eta

in Mechanical via

\dfrac{\omega_2}{\omega_1}=\dfrac{N_1}{N_2},\quad \dfrac{T_2}{T_1}=\eta\dfrac{N_2}{N_1}

ω2/ω1 = N1/N2, T2/T1 = η N2/N1. Teeth count sets the reduction. Use it when a real mechanical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

N_1 = 18.000\,\mathrm{—}

N_2 = 54.000\,\mathrm{—}

\omega_1 = 120.000\,\mathrm{rad/s}

T_1 = 40.000\,\mathrm{N·m}

\eta = 0.970\,\mathrm{—}

, the governing relation

\dfrac{\omega_2}{\omega_1}=\dfrac{N_1}{N_2},\quad \dfrac{T_2}{T_1}=\eta\dfrac{N_2}{N_1}

yields

\omega_2 = 40.00\,\mathrm{rad/s}

T_2 = 116.4\,\mathrm{N·m}

. Two pitch circles, a ratio arrow, a torque pair. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Pinion teeth N_118 —Wheel teeth N_254 —Pinion speed \omega_1120 rad/sPinion torque T_140 N·mMesh efficiency \eta0.97 —

Outputs

Wheel speed \omega_240.00 rad/s
Wheel torque T_2116.4 N·m

Reading speed

Watch on YouTube

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Narration of this film

Two pitch circles, a ratio arrow, a torque pair.

A meshing constraint: arc speeds match at the pitch point. Efficiency η < 1 accounts for tooth friction.

Reading speed

Watch on YouTube