MEC-29

Flywheel fluctuation energy

ΔE = I ω² Cs with Cs = (ωmax−ωmin)/ω. Stored kinetic buffer.

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EnergyShigley

Governing equation

\Delta E=I\omega^2 C_s

$\omega \in \left[10, 600\right]$

where

I: Inertia (kg·m²)
\omega: Mean speed (rad/s)
C_s: Fluctuation coeff. (—)
\Delta E: Energy fluctuation (kJ)

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

\Delta E=I\omega^2 C_s

$\omega \in \left[10, 600\right]$

$\omega = 10$ $\Delta E = 0.02$

Along the curve

②

Domain of the variable — blue

\omega_{\min} = 10,\quad \omega_{\max} = 600

$\omega \in \left[10, 600\right]$

$\omega = 10$ $\omega = 10$

Along the curve

③

Response curve — green

\Delta E=I\omega^2 C_s

$\omega \in \left[10, 600\right]$

$\omega = 10$ $\Delta E = 0.02$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\Delta E}{\mathrm{d}\omega}

$\omega \in \left[10, 600\right]$

$\omega = 13.71$ $\Delta E^{\prime} = 0.005$

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-29 — Flywheel fluctuation energy) is the form associated with Shigley. Working symbols: $I$ , $\omega$ , $C_s$ $\rightarrow$ $\Delta E$ . A flywheel stores ½ I ω². The energy exchanged while speed fluctuates by Cs is I ω² Cs.

Purpose

Purpose: compute

\Delta E

from

I

\omega

C_s

in Mechanical via

\Delta E=I\omega^2 C_s

ΔE = I ω² Cs with Cs = (ωmax−ωmin)/ω. Stored kinetic buffer. Use it when a real mechanical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

I = 4.000\,\mathrm{kg·m^{2}}

\omega = 150.000\,\mathrm{rad/s}

C_s = 0.050\,\mathrm{—}

, the governing relation

\Delta E=I\omega^2 C_s

yields

\Delta E = 4.500\,\mathrm{kJ}

. A disk, an ω band, an energy pulse. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Inertia I4 kg·m²Mean speed \omega150 rad/sFluctuation coeff. C_s0.05 —

Outputs

Energy fluctuation \Delta E4.500 kJ

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Narration of this film

A disk, an ω band, an energy pulse.

A flywheel stores ½ I ω². The energy exchanged while speed fluctuates by Cs is I ω² Cs.

Reading speed

Watch on YouTube