Quote:
Originally Posted by Tom Line
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The link above is a nice explanation.
Concisely, the square in the formula for reflected inertia comes from one simple observation: when frictional losses in the gearing are neglected, kinetic energy must be the same when observed on either the high-speed shaft or the low-speed shaft.
For a rotating system, KE = 0.5 * J * w^2, where J is moment of inertia and w is angular velocity.
So to make observed KE equal on two shafts coupled through a geartrain with speed ratio (w2/w1), J1/J2 must equal (w2/w1)^2.
[Using SI units we would express w in rad/sec, J in kg-m^2 or Joule-sec^2, and KE in Joule.]
As an example: a 50cm diameter steel flywheel 10 cm thick has about 5 Joule-sec^2 moment of inertia. When it spins at one rad/sec its kinetic energy is about 2.5 Joule. If we couple this flywheel to a 10:1 gearbox the shaft on the other side will spin at 10 rad/sec, the kinetic energy measured on the high-speed shaft will still be about 2.5 Joule, and the inertia of the flywheel reflected to the high-speed shaft will be about 0.05 Joule-sec^2.
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Richard Wallace
Mentor since 2011 for FRC 3620 Average Joes (St. Joseph, Michigan)
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since 2003
I believe in intuition and inspiration. Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.
(Cosmic Religion : With Other Opinions and Aphorisms (1931) by Albert Einstein, p. 97)