reflected inertia

How can I explain why a gear ratio reduces the inertia by the gear ratio squared? I do not understand why it is squared.


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.

Thanks to both of you. Your responses helped greatly.